Mathematics Matters Final Report

Mathematics Matters Final Report

www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching.

CONTENTS

Executive summary

2

Background

2

Outcomes

2

Concluding remarks

6

Rationale

7

Process

8

Outcomes

9

Values

9

Types of learning outcome and classroom activities implied by these values

10

Comparing ideal values with values in practice

11

Illustrating different types of learning outcome

12

Principles that guide teaching

18

Unhelpful principles

21

Obstacles to progress

21

Strategies for overcoming obstacles

24

Appendices

27

Appendix 1 Questionnaire used in colloquia and for online responses

27

Appendix 2 List of participants

34

The following additional information is available on the NCETM portal at www.ncetm.org.uk. This may be of interest to those who would like to learn more about the background to the study, the methodology employed and the tools provided to and by participants. quotations •SomeStimulus extracts from writing over the last 100 years that describe characteristics of learning and teaching mathematics. (Sent out with the invitation to the initial conference in order to set an historical context.) Lesson accounts •Fifty-seven lesson accounts received at the colloquia, all of which have been checked and validated. Acknowledgement The National Centre for Excellence in the Teaching of Mathematics (NCETM) is greatly indebted to Dr. Malcolm Swan, University of Nottingham, who chaired the project, devised the activities and co-wrote this report.

paper presented at the introductory conference •WhatDiscussion constitutes the effective learning of mathematics? A discussion paper for the NCETM by Malcolm Swan

•

A summary of the report

The direct involvement and leadership of Peter Lacey and Sarah Mann (ECARDA Ltd.), in project management, organisation of events and collection and analysis of data, also contributed greatly to the completion of the final report. We wish to express our gratitude to the many mathematics teachers and other stakeholders, listed in Appendix 2, who contributed their time, views, and experiences. Our thanks also go to the NCETM advisory group and others who contributed to the critical review of the report. 1

CONTENTS

Executive summary

2

Background

2

Outcomes

2

Concluding remarks

6

Rationale

7

Process

8

Outcomes

9

Values

9


10

Comparing ideal values with values in practice

11

Illustrating different types of learning outcome

12

Principles that guide teaching

18


21


21


24

Appendices

27


27

Appendix 2 List of participants

34

The following additional information is available on the NCETM portal at www.ncetm.org.uk. This may be of interest to those who would like to learn more about the background to the study, the methodology employed and the tools provided to and by participants. quotations •SomeStimulus extracts from writing over the last 100 years that describe characteristics of learning and teaching mathematics. (Sent out with the invitation to the initial conference in order to set an historical context.) Lesson accounts •Fifty-seven lesson accounts received at the colloquia, all of which have been checked and validated. Acknowledgement The National Centre for Excellence in the Teaching of Mathematics (NCETM) is greatly indebted to Dr. Malcolm Swan, University of Nottingham, who chaired the project, devised the activities and co-wrote this report.

paper presented at the introductory conference •WhatDiscussion constitutes the effective learning of mathematics? A discussion paper for the NCETM by Malcolm Swan

•

A summary of the report

The direct involvement and leadership of Peter Lacey and Sarah Mann (ECARDA Ltd.), in project management, organisation of events and collection and analysis of data, also contributed greatly to the completion of the final report. We wish to express our gratitude to the many mathematics teachers and other stakeholders, listed in Appendix 2, who contributed their time, views, and experiences. Our thanks also go to the NCETM advisory group and others who contributed to the critical review of the report. 1

EXECUTIVE SUMMARY

Background

While recognising that this is only the beginning of a national consultation, the findings suggest that there are surprisingly high levels of agreement with the values and principles that are set out below. Much of what follows may sound like ‘peace, motherhood and apple pie’, but its implications are far-reaching. Also, while noting that the views expressed in this report are those of the participants alone, the NCETM fully endorses the values and principles agreed. The NCETM therefore looks forward to this report stimulating further debate and providing a basis for a wider consensus among the community of mathematics educators, assessment designers and policy makers in this country.

Over the past year, the National Centre for Excellence in the Teaching of Mathematics (NCETM) has undertaken a consultation, Mathematics Matters, to review and describe the values and practices considered to be most important and effective by the mathematics education community. The purpose of the consultation is:

•

to establish an agreed set of valued learning outcomes and an agreed set of principles, grounded in evidence, that underpins the learning and teaching of mathematics

•

to produce a collection of lesson accounts from a range of different settings in order to illustrate what the values and principles might look like in the classroom

•

to ascertain the relative values put onto the different learning outcomes in an ideal case and to compare these with perceptions of relative values in practice

•

to identify any barriers to the values and principles being translated into practice, and to propose strategies for overcoming them.

Outcomes

Types of learning activity implied

Fluency in recalling facts and performing skills

• •

Memorising names and notations Practising algorithms and procedures for fluency and ‘mastery’

Conceptual understanding and interpretations for representations

• • • •

Discriminating between examples and nonexamples of concepts Generating representations of concepts Constructing networks of relationships between mathematical concepts Interpreting and translating between representations of concepts

Strategies for investigation and problem solving

• • • •

Formulating situations and problems for investigation Constructing, sharing, refining, and comparing strategies for exploration and solution Monitoring one’s own progress during problem solving and investigation Interpreting, evaluating solutions and communicating results

Awareness of the nature and values of the educational system

• • •

Recognising different purposes of learning mathematics Developing appropriate strategies for learning/ reviewing mathematics Appreciating aspects of performance valued by the examination system

Appreciation of the power of mathematics in society

• • • •

Appreciating mathematics as human creativity (plus historical aspects) Creating and critiquing ‘mathematical models’ of situations Appreciating uses/abuses of mathematics in social contexts Using mathematics to gain power over problems in one’s own life

1. Valuing what is important Participants reiterated the central importance of mathematics in the curriculum, in particular citing its social, personal and intrinsic worth. This endorsed the values expressed in the Cockcroft (1982) and Smith (2004) enquiries, and the National Curriculum (1999, 2007). They recognised that teaching should value each of the following learning outcomes though with a rather different emphasis to that currently seen in most classrooms:

A draft set of values and principles for teaching and learning were established at an initial conference in London, in May 2007. These were then discussed, amended and exemplified through lesson accounts at a series of regional colloquia over the following year. A first progress report was issued by the NCETM immediately after the initial conference and a second progress report was issued in December 2007. This is the final report on the first two phases. Full versions of all the reports, the stimulus materials used, and a database of responses and lesson accounts may be found on the NCETM portal1. To date, we have consulted with over 150 mathematics educators, with representation from early years and primary teachers, secondary teachers, adult education teachers and teacher educators.

• •

fluency in recalling facts and performing skills conceptual understanding and interpretations for representations

• •

strategies for investigation and problem solving

•

appreciation of the power of mathematics in society.

awareness of the nature and values of the educational system

Participants agreed that, in order to achieve these different learning outcomes, different classroom activities and teaching methods need to be deployed. These are described in the table, right.

Participants have echoed Cockcroft (1982, para 242) in recognising that it is not possible to define a single 'best practice' in mathematics teaching. There are many different types of learning, and a wide range of teaching methods will need to be deployed, appropriate to the learners and the particular learning outcomes desired. Mathematics Matters attempted to clarify the types of learning that the mathematics community values most highly and identify underlying principles that might be helpful in guiding the choice of appropriate teaching methods. It also sought to identify common reasons why teachers do not always choose appropriate methods.

1

Learning outcomes sought

When participants were asked to compare their “vision for an ideal mathematics curriculum” with the values that are implied by the “curriculum that is currently implemented in most schools, colleges and other settings”, a clear pattern emerged. Participants consistently reported that:

•

too much time is spent developing “fluency in recalling facts and performing skills” to the detriment of other aspects

•

much greater emphasis should be placed on the remaining four learning outcomes, with particular emphasis being placed on “conceptual understanding and interpretations for representations” and “strategies for investigation and problem solving”.

There was remarkably little variation in these views either across phases of education or between the geographical areas where meetings were held.

http://www.ncetm.org.uk

2

3

EXECUTIVE SUMMARY

Background

While recognising that this is only the beginning of a national consultation, the findings suggest that there are surprisingly high levels of agreement with the values and principles that are set out below. Much of what follows may sound like ‘peace, motherhood and apple pie’, but its implications are far-reaching. Also, while noting that the views expressed in this report are those of the participants alone, the NCETM fully endorses the values and principles agreed. The NCETM therefore looks forward to this report stimulating further debate and providing a basis for a wider consensus among the community of mathematics educators, assessment designers and policy makers in this country.

Over the past year, the National Centre for Excellence in the Teaching of Mathematics (NCETM) has undertaken a consultation, Mathematics Matters, to review and describe the values and practices considered to be most important and effective by the mathematics education community. The purpose of the consultation is:

•

to establish an agreed set of valued learning outcomes and an agreed set of principles, grounded in evidence, that underpins the learning and teaching of mathematics

•

to produce a collection of lesson accounts from a range of different settings in order to illustrate what the values and principles might look like in the classroom

•

to ascertain the relative values put onto the different learning outcomes in an ideal case and to compare these with perceptions of relative values in practice

•

to identify any barriers to the values and principles being translated into practice, and to propose strategies for overcoming them.

Outcomes



• •



• • • •



• • • •



• • •



• • • •


1. Valuing what is important Participants reiterated the central importance of mathematics in the curriculum, in particular citing its social, personal and intrinsic worth. This endorsed the values expressed in the Cockcroft (1982) and Smith (2004) enquiries, and the National Curriculum (1999, 2007). They recognised that teaching should value each of the following learning outcomes though with a rather different emphasis to that currently seen in most classrooms:

A draft set of values and principles for teaching and learning were established at an initial conference in London, in May 2007. These were then discussed, amended and exemplified through lesson accounts at a series of regional colloquia over the following year. A first progress report was issued by the NCETM immediately after the initial conference and a second progress report was issued in December 2007. This is the final report on the first two phases. Full versions of all the reports, the stimulus materials used, and a database of responses and lesson accounts may be found on the NCETM portal1. To date, we have consulted with over 150 mathematics educators, with representation from early years and primary teachers, secondary teachers, adult education teachers and teacher educators.

• •

fluency in recalling facts and performing skills conceptual understanding and interpretations for representations

• •

strategies for investigation and problem solving

•

appreciation of the power of mathematics in society.

awareness of the nature and values of the educational system

Participants agreed that, in order to achieve these different learning outcomes, different classroom activities and teaching methods need to be deployed. These are described in the table, right.

Participants have echoed Cockcroft (1982, para 242) in recognising that it is not possible to define a single 'best practice' in mathematics teaching. There are many different types of learning, and a wide range of teaching methods will need to be deployed, appropriate to the learners and the particular learning outcomes desired. Mathematics Matters attempted to clarify the types of learning that the mathematics community values most highly and identify underlying principles that might be helpful in guiding the choice of appropriate teaching methods. It also sought to identify common reasons why teachers do not always choose appropriate methods.

1


When participants were asked to compare their “vision for an ideal mathematics curriculum” with the values that are implied by the “curriculum that is currently implemented in most schools, colleges and other settings”, a clear pattern emerged. Participants consistently reported that:

•

too much time is spent developing “fluency in recalling facts and performing skills” to the detriment of other aspects

•

much greater emphasis should be placed on the remaining four learning outcomes, with particular emphasis being placed on “conceptual understanding and interpretations for representations” and “strategies for investigation and problem solving”.

There was remarkably little variation in these views either across phases of education or between the geographical areas where meetings were held.

http://www.ncetm.org.uk

2

3

•

2. Principles for teaching Participants suggested amendments to the draft set of principles established at the initial conference and a final version is set out below. These principles have a sound basis in research. Teaching is more effective when it:

• •

Builds on the knowledge learners already have Developing formative assessment techniques and adapting our teaching to accommodate individual learning needs.

•

Exposes and discusses common misconceptions and other surprising phenomena Using learning activities that expose current thinking, create ‘tensions’ by confronting learners with inconsistencies and surprises, and allow opportunities for resolution through discussion.

•

Uses higher-order questions Questioning is more effective in promoting mathematical thinking when it promotes explanation, application and synthesis rather than mere recall.

•

Makes appropriate use of whole-class interactive teaching, individual work and cooperative small group work Collaborative group work is more effective after learners have been given an opportunity for individual reflection. Activities are more effective when they encourage critical, constructive discussion, rather than uncritical acceptance or argument for its own sake. Shared goals and group accountability are important.

•

Encourages reasoning rather than ‘answer getting’ Often learners are more concerned with what they have ‘done’ than with what they have learned. It is better to aim for depth than for superficial ‘coverage’.

•

Uses rich, collaborative tasks Tasks which promote mathematical learning are accessible, extendable, encourage decision-making, promote discussion, encourage creativity, encourage ‘what if’ and ‘what if not’ questions.

•

Creates connections between topics both within and beyond mathematics and with the real world Learners often find it difficult to generalise and transfer their learning to other topics and contexts. Related concepts (such as division, fraction and ratio) remain unconnected. Effective teachers build bridges between ideas.

•

•

Uses resources, including technology, in creative and appropriate ways There are many rich resources that promote mathematical learning. For example, ICT offers new ways to engage with mathematics. At its best it is dynamic and visual: relationships become more tangible. ICT can provide feedback on actions and enhance interactivity and learner autonomy. Through its connectivity, ICT offers the means to access and share resources and – even more powerfully – the means by which learners can share their ideas within and across classrooms. Confronts difficulties rather than seeks to avoid or pre-empt them Effective teaching challenges learners and has high expectations of them. It does not seek to ‘smooth the path’ but creates realistic obstacles to be overcome. Confidence, persistence and learning are not attained solely through repeating successes, but by struggling with difficulties. Develops mathematical language through communicative activities Mathematics is a language that enables us to describe and model situations, think logically, frame and sustain arguments and communicate ideas with precision. Learners do not know mathematics until they can ‘speak’ it. Effective teaching therefore focuses on the communicative aspects of mathematics by developing oral and written mathematical language. Recognises both what has been learned and also how it has been learned What is to be learned cannot always be stated prior to the learning experience. After a learning event, however, it is important to reflect on the learning that has taken place, making this as explicit and memorable as possible. Effective teachers will also reflect on the ways in which learning has taken place, so that learners develop their own capacity to learn.

In Mathematics Matters, the NCETM invited participants to describe sample, illustrative lessons of ‘best practice’. These may be found on the NCETM portal and serve to ‘flesh out’ these general principles.

4

3. Obstacles to progress

4. Suggested ways forward

Participants identified many reasons for discrepancies between the current curriculum as implemented in schools and colleges and the curriculum they would wish to see. The most common reasons cited were:

Participants were also asked to suggest ways in which the above obstacles may be overcome. Recommendations centred around four themes, namely:

•

Society’s attitude towards mathematics Society and its media still promote mathematics as a ‘geeky’ subject and mathematical incompetence remains socially acceptable.

•

Teachers’ subject and pedagogical subject knowledge Many teachers lack confidence in the subject and an awareness and understanding of appropriate approaches and resources (including ICT). They also lack time for continuing professional development.

•

A taught curriculum defined by assessment Assessment should be defined by the curriculum aims, rather than the taught curriculum being defined by assessment. Teaching towards national tests and public examinations has led to an overemphasis on ‘covering content’ in a superficial manner, at the expense of developing deeper understanding and non-routine problem solving. This is, at least partly, due to the fragmented and artificial nature of many of these assessments.

•

•

The style and quality of textbooks and other resources There is still an over-reliance on pre-packaged schemes in all phases. Many of these, particularly those targeting specific assessments, do not promote the types of learning educators value. Initiatives that appear conflicting, disempowering and prescriptive A simplistic interpretation of national strategies and the apparent inflexibility and mechanistic nature of inspection regimes (particularly internal ones), can lead to the production of externally acceptable forms of behaviour (e.g. 3-part lessons, learning objectives written on the board before the lesson) and inhibit principled, imaginative teaching. Teachers report that inconsistencies often appear between the practices sought by inspections internal to their organisations (often by non-specialists) and those that are recommended by outside agencies.

5

•

Improve the provision and quality of professional development opportunities Participants proposed that continuing professional development should become an entitlement and expectation for all teachers. Effective, collaborative professional development should focus both on developing participants’ subject and pedagogical knowledge, rather than on the mechanics of implementing the latest ‘initiative’. Professional development should be characterised as reflective teachers researching their own practice and engaging with the research of others. Resources should be found to foster partnerships between institutions, particularly with those in HE.

•

Develop and share experience and resources for learning Participants saw a need for the NCETM and others to begin to collate and exemplify the combined experience, wisdom and expertise of teachers of mathematics, so that these may be shared more widely. A range of resources were suggested, including: mathematical tasks and activities that encourage collaborative work and the development of mathematical processes; accounts and authentic videos of lessons that exhibit a wide range of effective practices.

•

Use professional standards to inform others about the teaching and learning of mathematics Participants recommended that professional standards that describe and exemplify our values, principles and practices are developed and used to educate the perceptions of managers in schools and colleges so that more time may be made available for professional development and more informed performance assessment might take place. In addition, participants suggested that all those in mathematics education should continue to seek ways of promoting positive images of mathematics and mathematicians to society in general.

•

2. Principles for teaching Participants suggested amendments to the draft set of principles established at the initial conference and a final version is set out below. These principles have a sound basis in research. Teaching is more effective when it:

• •

Builds on the knowledge learners already have Developing formative assessment techniques and adapting our teaching to accommodate individual learning needs.

•

Exposes and discusses common misconceptions and other surprising phenomena Using learning activities that expose current thinking, create ‘tensions’ by confronting learners with inconsistencies and surprises, and allow opportunities for resolution through discussion.

•

Uses higher-order questions Questioning is more effective in promoting mathematical thinking when it promotes explanation, application and synthesis rather than mere recall.

•

Makes appropriate use of whole-class interactive teaching, individual work and cooperative small group work Collaborative group work is more effective after learners have been given an opportunity for individual reflection. Activities are more effective when they encourage critical, constructive discussion, rather than uncritical acceptance or argument for its own sake. Shared goals and group accountability are important.

•

Encourages reasoning rather than ‘answer getting’ Often learners are more concerned with what they have ‘done’ than with what they have learned. It is better to aim for depth than for superficial ‘coverage’.

•

Uses rich, collaborative tasks Tasks which promote mathematical learning are accessible, extendable, encourage decision-making, promote discussion, encourage creativity, encourage ‘what if’ and ‘what if not’ questions.

•

Creates connections between topics both within and beyond mathematics and with the real world Learners often find it difficult to generalise and transfer their learning to other topics and contexts. Related concepts (such as division, fraction and ratio) remain unconnected. Effective teachers build bridges between ideas.

•

•

Uses resources, including technology, in creative and appropriate ways There are many rich resources that promote mathematical learning. For example, ICT offers new ways to engage with mathematics. At its best it is dynamic and visual: relationships become more tangible. ICT can provide feedback on actions and enhance interactivity and learner autonomy. Through its connectivity, ICT offers the means to access and share resources and – even more powerfully – the means by which learners can share their ideas within and across classrooms. Confronts difficulties rather than seeks to avoid or pre-empt them Effective teaching challenges learners and has high expectations of them. It does not seek to ‘smooth the path’ but creates realistic obstacles to be overcome. Confidence, persistence and learning are not attained solely through repeating successes, but by struggling with difficulties. Develops mathematical language through communicative activities Mathematics is a language that enables us to describe and model situations, think logically, frame and sustain arguments and communicate ideas with precision. Learners do not know mathematics until they can ‘speak’ it. Effective teaching therefore focuses on the communicative aspects of mathematics by developing oral and written mathematical language. Recognises both what has been learned and also how it has been learned What is to be learned cannot always be stated prior to the learning experience. After a learning event, however, it is important to reflect on the learning that has taken place, making this as explicit and memorable as possible. Effective teachers will also reflect on the ways in which learning has taken place, so that learners develop their own capacity to learn.

In Mathematics Matters, the NCETM invited participants to describe sample, illustrative lessons of ‘best practice’. These may be found on the NCETM portal and serve to ‘flesh out’ these general principles.

4

3. Obstacles to progress

4. Suggested ways forward

Participants identified many reasons for discrepancies between the current curriculum as implemented in schools and colleges and the curriculum they would wish to see. The most common reasons cited were:

Participants were also asked to suggest ways in which the above obstacles may be overcome. Recommendations centred around four themes, namely:

•

Society’s attitude towards mathematics Society and its media still promote mathematics as a ‘geeky’ subject and mathematical incompetence remains socially acceptable.

•

Teachers’ subject and pedagogical subject knowledge Many teachers lack confidence in the subject and an awareness and understanding of appropriate approaches and resources (including ICT). They also lack time for continuing professional development.

•

A taught curriculum defined by assessment Assessment should be defined by the curriculum aims, rather than the taught curriculum being defined by assessment. Teaching towards national tests and public examinations has led to an overemphasis on ‘covering content’ in a superficial manner, at the expense of developing deeper understanding and non-routine problem solving. This is, at least partly, due to the fragmented and artificial nature of many of these assessments.

•

•

The style and quality of textbooks and other resources There is still an over-reliance on pre-packaged schemes in all phases. Many of these, particularly those targeting specific assessments, do not promote the types of learning educators value. Initiatives that appear conflicting, disempowering and prescriptive A simplistic interpretation of national strategies and the apparent inflexibility and mechanistic nature of inspection regimes (particularly internal ones), can lead to the production of externally acceptable forms of behaviour (e.g. 3-part lessons, learning objectives written on the board before the lesson) and inhibit principled, imaginative teaching. Teachers report that inconsistencies often appear between the practices sought by inspections internal to their organisations (often by non-specialists) and those that are recommended by outside agencies.

5

•

Improve the provision and quality of professional development opportunities Participants proposed that continuing professional development should become an entitlement and expectation for all teachers. Effective, collaborative professional development should focus both on developing participants’ subject and pedagogical knowledge, rather than on the mechanics of implementing the latest ‘initiative’. Professional development should be characterised as reflective teachers researching their own practice and engaging with the research of others. Resources should be found to foster partnerships between institutions, particularly with those in HE.

•

Develop and share experience and resources for learning Participants saw a need for the NCETM and others to begin to collate and exemplify the combined experience, wisdom and expertise of teachers of mathematics, so that these may be shared more widely. A range of resources were suggested, including: mathematical tasks and activities that encourage collaborative work and the development of mathematical processes; accounts and authentic videos of lessons that exhibit a wide range of effective practices.

•

Use professional standards to inform others about the teaching and learning of mathematics Participants recommended that professional standards that describe and exemplify our values, principles and practices are developed and used to educate the perceptions of managers in schools and colleges so that more time may be made available for professional development and more informed performance assessment might take place. In addition, participants suggested that all those in mathematics education should continue to seek ways of promoting positive images of mathematics and mathematicians to society in general.

RATIONALE

•

Influence the extent and quality of key stage assessments and public examinations Participants generally felt that they wanted to see the elimination of, or at least a reduction in, the quantity of end of key-stage external assessment’ and so create more time for teaching and learning, including formative assessment. They also felt that the content of these assessments should be revised to reflect the broader aims of mathematics teaching. Participants also felt that the ‘high stakes’ basis of league tables should be altered so that competition between schools is replaced by collaboration among teachers.

It was 20 years since the publication of ‘Better Mathematics’ (Ahmed, 1987) and 25 years since the publication of ‘Mathematics Counts’ (Cockcroft, 1982). Each of these documents articulated views on what constitutes the effective learning of mathematics – informed by accumulated research findings and interpreted through the prevailing culture and values. For example, in ‘Better Mathematics’ there is the memorable Statement 4:

The NCETM will share all the underpinning evidence in the report with partners and stakeholders and will explore in greater depth the messages in the first three themes and use them to inform and influence its work.

The NCETM framed the key question, “What constitutes the effective learning of mathematics?” on the understanding that the answers to this question are fundamental. Methods of teaching, the design of the curriculum, the use of assessment (both formative and summative), the organisation of learning groups, the selection and use of supporting resources and materials, the initial and continued professional development of teachers are all consequent on the answers to this question. It is appropriate that the question is asked by each generation and the answers refreshed in the light of experience and the latest research findings. Equally, the answers need to be interpreted through the prevailing culture and values.

Mathematics is effectively learned only by experimenting, questioning, reflecting, discovering, inventing and discussing. Thus, for children, mathematics should be a kind of learning which requires a minimum of factual knowledge and a great deal of experience in dealing with situations using particular kinds of thinking skills.

Concluding remarks Mathematics Matters is only a beginning. It is not suggested that the values and principles set out are in any way complete or exhaustive. Indeed, the NCETM intends that they will stimulate a wider, more informed debate. Further exemplification is essential if they are not to become just another collection of 'worthy words'.

And in ‘Mathematics Counts’ is found: 242 We wish now to discuss the implications of the previous sections for work in the classroom. We are aware that there are some teachers who would wish us to indicate a definitive style of teaching mathematics, but we do not believe that this is either desirable or possible. Approaches to the teaching of a particular piece of mathematics need to be related to the topic itself and to the abilities of both teachers and pupils. Because of the differences of personality and circumstance, methods which may be extremely successful with one teacher and one group of pupils will not necessarily be suitable for use by another teacher or with a different group of pupils. Nevertheless, we believe that there are certain elements which need to be present in successful mathematics teaching to pupils of all ages.

The process has proved to be most stimulating and has offered a valuable professional development experience for many of those participating. In particular, the educators involved have reported how much they have valued the sharing of lesson accounts and reflecting on the values and principles these reveal. The National Centre therefore intends that this process will continue. To date, the National Centre has collected 57 lesson accounts to illustrate the wide range of practices that illustrate the values and principles stated above. These accounts are too extensive to include in this report so they are available on the NCETM portal, categorised by:

• • •

In setting out this list we are aware that we are not saying anything which has not already been said many times over many years. The list which we have given has appeared, by implication if not explicitly, in official reports, DES publications, HMI discussion papers and the journals and publications of the professional mathematics associations. Yet we are aware that although there are some classrooms in which the teaching includes, as a matter of course, all the elements we have listed, there are still many in which the mathematics teaching does not include even a majority of these elements.

In 2007, the mathematics community celebrated two significant anniversaries.

Thus, the NCETM, with its clear brief to provide coordination and leadership on all aspects of professional development for teachers of mathematics, took responsibility for stimulating and undertaking a national debate in order to:

243 Mathematics teaching at all levels should include opportunities for

• • • • • •

their underlying values the principles they illustrate the nature of the learner group.

The values and principles set out above are intended to encourage all those involved in mathematics education to be more explicit about their own priorities and goals and the types of task and classroom activity that they would wish to see. We strongly believe that it is possible for learners at every level to become excited about mathematics and about being mathematicians, particularly when they begin to experience the broader range of activities that are proposed.

6

exposition by the teacher; discussion between teacher and pupils and between pupils themselves; appropriate practical work; consolidation and practice of fundamental skills and routines; problem solving, including the application of mathematics to everyday situations; investigational work.

7

•

establish an agreed set of valued learning outcomes and an agreed, and evidence-based, set of principles that underpins the teaching and learning of mathematics

•

produce a collection of lesson accounts from a range of different settings in order to illustrate what the values and principles might look like in the classroom

•

ascertain the relative emphasis put onto the different valued learning outcomes in an ideal case and to compare these with perceptions of relative emphasis in practice, and

•

identify any barriers to the values and principles being translated into practice, and to propose strategies for overcoming them.

RATIONALE

•

Influence the extent and quality of key stage assessments and public examinations Participants generally felt that they wanted to see the elimination of, or at least a reduction in, the quantity of end of key-stage external assessment’ and so create more time for teaching and learning, including formative assessment. They also felt that the content of these assessments should be revised to reflect the broader aims of mathematics teaching. Participants also felt that the ‘high stakes’ basis of league tables should be altered so that competition between schools is replaced by collaboration among teachers.

It was 20 years since the publication of ‘Better Mathematics’ (Ahmed, 1987) and 25 years since the publication of ‘Mathematics Counts’ (Cockcroft, 1982). Each of these documents articulated views on what constitutes the effective learning of mathematics – informed by accumulated research findings and interpreted through the prevailing culture and values. For example, in ‘Better Mathematics’ there is the memorable Statement 4:


The NCETM framed the key question, “What constitutes the effective learning of mathematics?” on the understanding that the answers to this question are fundamental. Methods of teaching, the design of the curriculum, the use of assessment (both formative and summative), the organisation of learning groups, the selection and use of supporting resources and materials, the initial and continued professional development of teachers are all consequent on the answers to this question. It is appropriate that the question is asked by each generation and the answers refreshed in the light of experience and the latest research findings. Equally, the answers need to be interpreted through the prevailing culture and values.

Mathematics is effectively learned only by experimenting, questioning, reflecting, discovering, inventing and discussing. Thus, for children, mathematics should be a kind of learning which requires a minimum of factual knowledge and a great deal of experience in dealing with situations using particular kinds of thinking skills.

Concluding remarks Mathematics Matters is only a beginning. It is not suggested that the values and principles set out are in any way complete or exhaustive. Indeed, the NCETM intends that they will stimulate a wider, more informed debate. Further exemplification is essential if they are not to become just another collection of 'worthy words'.

And in ‘Mathematics Counts’ is found: 242 We wish now to discuss the implications of the previous sections for work in the classroom. We are aware that there are some teachers who would wish us to indicate a definitive style of teaching mathematics, but we do not believe that this is either desirable or possible. Approaches to the teaching of a particular piece of mathematics need to be related to the topic itself and to the abilities of both teachers and pupils. Because of the differences of personality and circumstance, methods which may be extremely successful with one teacher and one group of pupils will not necessarily be suitable for use by another teacher or with a different group of pupils. Nevertheless, we believe that there are certain elements which need to be present in successful mathematics teaching to pupils of all ages.

The process has proved to be most stimulating and has offered a valuable professional development experience for many of those participating. In particular, the educators involved have reported how much they have valued the sharing of lesson accounts and reflecting on the values and principles these reveal. The National Centre therefore intends that this process will continue. To date, the National Centre has collected 57 lesson accounts to illustrate the wide range of practices that illustrate the values and principles stated above. These accounts are too extensive to include in this report so they are available on the NCETM portal, categorised by:

• • •

In setting out this list we are aware that we are not saying anything which has not already been said many times over many years. The list which we have given has appeared, by implication if not explicitly, in official reports, DES publications, HMI discussion papers and the journals and publications of the professional mathematics associations. Yet we are aware that although there are some classrooms in which the teaching includes, as a matter of course, all the elements we have listed, there are still many in which the mathematics teaching does not include even a majority of these elements.

In 2007, the mathematics community celebrated two significant anniversaries.

Thus, the NCETM, with its clear brief to provide coordination and leadership on all aspects of professional development for teachers of mathematics, took responsibility for stimulating and undertaking a national debate in order to:

243 Mathematics teaching at all levels should include opportunities for

• • • • • •

their underlying values the principles they illustrate the nature of the learner group.

The values and principles set out above are intended to encourage all those involved in mathematics education to be more explicit about their own priorities and goals and the types of task and classroom activity that they would wish to see. We strongly believe that it is possible for learners at every level to become excited about mathematics and about being mathematicians, particularly when they begin to experience the broader range of activities that are proposed.

6

exposition by the teacher; discussion between teacher and pupils and between pupils themselves; appropriate practical work; consolidation and practice of fundamental skills and routines; problem solving, including the application of mathematics to everyday situations; investigational work.

7

•

establish an agreed set of valued learning outcomes and an agreed, and evidence-based, set of principles that underpins the teaching and learning of mathematics

•

produce a collection of lesson accounts from a range of different settings in order to illustrate what the values and principles might look like in the classroom

•

ascertain the relative emphasis put onto the different valued learning outcomes in an ideal case and to compare these with perceptions of relative emphasis in practice, and

•

identify any barriers to the values and principles being translated into practice, and to propose strategies for overcoming them.

PROCESS

OUTCOMES

An initial conference, with 71 leading professionals in the field of mathematics education, was held in London in June 2007 to stimulate debate by:

(P.Williams)

•

identifying, confirming and agreeing values and principles that underpin the effective teaching and learning of mathematics

•

illustrating, through examples, how practice may reflect and interpret these core values and principles, and

•

Excellent – wonderful to discuss teaching etc. for the day. A pleasure to listen to colleagues.

the values and principles proposed at the initial conference, after which participants were invited to work together to respond in writing to these proposals, including their views on any differences between ideal and practised values. Next, participants returned to the lesson account, discussed it in depth with two other participants and then associated the account with the values and principles it best illustrated.

exploring the factors that inhibit or modify the implementation of these values and principles.

This was followed by a series of six one-day regional colloquia, held between September 07 and February 08 that were designed to:

•

test levels of agreement with the values and principles articulated at the initial conference and to amend and refine them as appropriate, and to

•

begin to build a collection of lesson accounts that illustrate what the values and principles may look like in practice.

The afternoon started by identifying obstacles to teaching mathematics in accordance with the values and principles that had been identified. This was followed by group sessions that proposed strategies to overcome these obstacles. Throughout the day, participants wrote their responses to all the activities on prepared templates (see Appendix 1), which facilitated subsequent collation and analyses. Participants had the option of having their lesson accounts ascribed or not.

These regional colloquia involved representation from early years and primary teachers, secondary teachers, teacher trainers and educators, adult education teachers (including prison), and the NCETM team and consultants. We recognise, of course, that this is not a representative sample, but rather an invited group of informed participants. In addition, we recognise the underrepresentation of early years and primary phase practitioners. The Centre has plans in place to address this matter.

Though the emphasis in each colloquium day was to gather information and views from participants, as opposed to provide information to them, their feedback overwhelmingly shows that participants found the day of personal, professional development value. They described time for reflection on their own practice and to discuss values and principles of mathematics education as among the most useful elements.

The table below shows the location and timing of each colloquium.

Location

Date

Leicester Bath Canterbury Newcastle Manchester Peterborough

28-09-07 18-10-07 02-11-07 11-12-07 25-01-08 07-02-08

It was really good to have time to reflect and think about my own practice and the values and principles that underpin my own teaching.

Attendees 13 11 14 11 27 7

(J.Knightbridge)

Values

Mathematics enables learners to participate in life both at work and at home.

The initial conference began by addressing the reasons why mathematics is considered to be important. Participants then tried to identify how these values reveal themselves in classroom practices. This, in turn, further led us all to reflect on how far current classroom practices are in accord with our values and possible reasons for any discrepancies.

(J.Back)

The personal value of mathematics Participants valued learners seeing themselves as mathematicians and appreciating the beauty of mathematics. They stressed the benefits of mental empowerment attained through cycles of abstraction and experience and the confidence gained when finding strategies to approach and solve problems. Mathematical learning was also felt to contribute to spiritual, moral and cultural development.

In both the initial conference and subsequent colloquia, there was broad agreement on why mathematics is important. Our values echoed those found in, for example, the Cockcroft (1982) and Smith (2004) reports, the National Curriculum Programmes of Study (QCA, 2007). Three foci were identified:

Mathematics has value in enabling groups and societies to function effectively and creatively.

Mathematics equips pupils with uniquely powerful ways to describe, analyse and change the world. It can stimulate moments of pleasure and wonder for all pupils when they solve a problem for the first time, discover a more elegant solution, or notice hidden connections. Pupils who are functional in mathematics and financially capable are able to think independently in applied and abstract ways, and can reason, solve problems and assess risk. (QCA 2007)

(R.Griffiths, P.Hough)

The social value of mathematics Participants felt that mathematics continues to make a substantial contribution to the effective and creative functioning of our increasingly technological society, by equipping learners with the life skills they need at work and at home. Furthermore, mathematics can enable learners to develop the social skills they need for effective collaboration with others and to become aware of and appreciate the different perspectives they may have. Learners may also begin to develop an awareness of the historical-cultural-social origins of mathematics and its role in modern society.

…learners seeing themselves as mathematicians. (S.Feller)

The intrinsic value of mathematics Participants valued mathematics as a creative discipline in its own right. They noted that mathematics offers learners a uniquely powerful set of tools with which they can interpret, organise and sort information, analyse and solve problems and communicate with others. Mathematics was also considered to be the ‘mother of all languages’, a grammatology of thinking and a way of making meanings, viewing and describing the world.

Mathematics is of central importance to modern society. It provides the language and analytical tools underpinning much of our scientific and industrial research and development. Mathematical concepts, models and techniques are also key to many vital areas of the knowledge economy, including the finance and ICT industries. Mathematics is crucially important too, for the employment opportunities and achievements of individual citizens. (Smith, 2004 foreword, page v)

Mathematics is a creative discipline. The language of mathematics is international. The subject transcends cultural boundaries and its importance is universally recognised. Mathematics has developed over time as a means of solving problems and also for its own sake. (QCA 2007)

Mathematical thinking is important for all members of a modern society as a habit of mind for its use in the workplace, business and finance; and for personal decision-making. Mathematics is fundamental to national prosperity in providing tools for understanding science, engineering, technology and economics. It is essential in public decision-making and for participation in the knowledge economy. (QCA, 2007)

Each colloquium day started with an invitation to each participant to spend half an hour writing an account of a memorable, inspirational mathematics lesson, either taught or observed. This was followed by a discussion on

8

…confidence in strategies to approach and solve problems. (J.Golding, B.Murphy) 9

PROCESS

OUTCOMES

An initial conference, with 71 leading professionals in the field of mathematics education, was held in London in June 2007 to stimulate debate by:

(P.Williams)

•

identifying, confirming and agreeing values and principles that underpin the effective teaching and learning of mathematics

•

illustrating, through examples, how practice may reflect and interpret these core values and principles, and

•

Excellent – wonderful to discuss teaching etc. for the day. A pleasure to listen to colleagues.

the values and principles proposed at the initial conference, after which participants were invited to work together to respond in writing to these proposals, including their views on any differences between ideal and practised values. Next, participants returned to the lesson account, discussed it in depth with two other participants and then associated the account with the values and principles it best illustrated.

exploring the factors that inhibit or modify the implementation of these values and principles.

This was followed by a series of six one-day regional colloquia, held between September 07 and February 08 that were designed to:

•

test levels of agreement with the values and principles articulated at the initial conference and to amend and refine them as appropriate, and to

•

begin to build a collection of lesson accounts that illustrate what the values and principles may look like in practice.

The afternoon started by identifying obstacles to teaching mathematics in accordance with the values and principles that had been identified. This was followed by group sessions that proposed strategies to overcome these obstacles. Throughout the day, participants wrote their responses to all the activities on prepared templates (see Appendix 1), which facilitated subsequent collation and analyses. Participants had the option of having their lesson accounts ascribed or not.

These regional colloquia involved representation from early years and primary teachers, secondary teachers, teacher trainers and educators, adult education teachers (including prison), and the NCETM team and consultants. We recognise, of course, that this is not a representative sample, but rather an invited group of informed participants. In addition, we recognise the underrepresentation of early years and primary phase practitioners. The Centre has plans in place to address this matter.

Though the emphasis in each colloquium day was to gather information and views from participants, as opposed to provide information to them, their feedback overwhelmingly shows that participants found the day of personal, professional development value. They described time for reflection on their own practice and to discuss values and principles of mathematics education as among the most useful elements.

The table below shows the location and timing of each colloquium.

Location

Date

Leicester Bath Canterbury Newcastle Manchester Peterborough

28-09-07 18-10-07 02-11-07 11-12-07 25-01-08 07-02-08

It was really good to have time to reflect and think about my own practice and the values and principles that underpin my own teaching.

Attendees 13 11 14 11 27 7

(J.Knightbridge)

Values

Mathematics enables learners to participate in life both at work and at home.

The initial conference began by addressing the reasons why mathematics is considered to be important. Participants then tried to identify how these values reveal themselves in classroom practices. This, in turn, further led us all to reflect on how far current classroom practices are in accord with our values and possible reasons for any discrepancies.

(J.Back)

The personal value of mathematics Participants valued learners seeing themselves as mathematicians and appreciating the beauty of mathematics. They stressed the benefits of mental empowerment attained through cycles of abstraction and experience and the confidence gained when finding strategies to approach and solve problems. Mathematical learning was also felt to contribute to spiritual, moral and cultural development.

In both the initial conference and subsequent colloquia, there was broad agreement on why mathematics is important. Our values echoed those found in, for example, the Cockcroft (1982) and Smith (2004) reports, the National Curriculum Programmes of Study (QCA, 2007). Three foci were identified:

Mathematics has value in enabling groups and societies to function effectively and creatively.

Mathematics equips pupils with uniquely powerful ways to describe, analyse and change the world. It can stimulate moments of pleasure and wonder for all pupils when they solve a problem for the first time, discover a more elegant solution, or notice hidden connections. Pupils who are functional in mathematics and financially capable are able to think independently in applied and abstract ways, and can reason, solve problems and assess risk. (QCA 2007)

(R.Griffiths, P.Hough)

The social value of mathematics Participants felt that mathematics continues to make a substantial contribution to the effective and creative functioning of our increasingly technological society, by equipping learners with the life skills they need at work and at home. Furthermore, mathematics can enable learners to develop the social skills they need for effective collaboration with others and to become aware of and appreciate the different perspectives they may have. Learners may also begin to develop an awareness of the historical-cultural-social origins of mathematics and its role in modern society.

…learners seeing themselves as mathematicians. (S.Feller)

The intrinsic value of mathematics Participants valued mathematics as a creative discipline in its own right. They noted that mathematics offers learners a uniquely powerful set of tools with which they can interpret, organise and sort information, analyse and solve problems and communicate with others. Mathematics was also considered to be the ‘mother of all languages’, a grammatology of thinking and a way of making meanings, viewing and describing the world.

Mathematics is of central importance to modern society. It provides the language and analytical tools underpinning much of our scientific and industrial research and development. Mathematical concepts, models and techniques are also key to many vital areas of the knowledge economy, including the finance and ICT industries. Mathematics is crucially important too, for the employment opportunities and achievements of individual citizens. (Smith, 2004 foreword, page v)

Mathematics is a creative discipline. The language of mathematics is international. The subject transcends cultural boundaries and its importance is universally recognised. Mathematics has developed over time as a means of solving problems and also for its own sake. (QCA 2007)

Mathematical thinking is important for all members of a modern society as a habit of mind for its use in the workplace, business and finance; and for personal decision-making. Mathematics is fundamental to national prosperity in providing tools for understanding science, engineering, technology and economics. It is essential in public decision-making and for participation in the knowledge economy. (QCA, 2007)

Each colloquium day started with an invitation to each participant to spend half an hour writing an account of a memorable, inspirational mathematics lesson, either taught or observed. This was followed by a discussion on

8

…confidence in strategies to approach and solve problems. (J.Golding, B.Murphy) 9

…to organise and sort information, to solve problems and explore the world.

At the initial conference there was also considerable agreement as to the type of learning outcomes valued and the different types of classroom activity that these outcomes might imply. The following non-hierarchical list was found helpful:

(R.Hyde)




• •



• • • •



• • • •


• • •


• • • •




Comparing ideal values with values in practice When participants were asked to compare their “vision for an ideal mathematics curriculum” with the values that are implied by the “curriculum that is currently implemented in most schools and other settings”, a consistent pattern emerged. Participants recorded on a four-point scale how frequently mathematics lessons should include each learning outcome and also how frequently mathematics lessons, in their view, actually do reflect each learning outcome (1 = hardly ever, 4 = almost every lesson). The aggregated responses of 133 initial conference and colloquia participants are shown in the graph below. The results show that participants consider that fluency is overvalued in the current curriculum, while the remaining values are undervalued. Perhaps surprisingly, this pattern of response was repeated consistently across responses from each phase of education (primary, secondary and tertiary) and across all regions. We note again, however, that due to the small and self-selecting nature of the sample involved (particularly with regard to primary teachers) these results may not reflect those of the profession as a whole. All respondents: Ideal and Current Outcomes – Weighted Average where high = most important

Ideal Curriculum Current Curriculum 4

N = 133

Source: (Swan, 2006 p.42)

3.5

3

Weighted Average


2.5

2

1.5

1

0.5

…a way of making meanings, of viewing and describing the world. (P.Lacey, P.Drake)

10

0 Fluency in recalling facts and performing skills

Interpretations for concepts and representations


11


Appreciation of the power of mathematics

…to organise and sort information, to solve problems and explore the world.

At the initial conference there was also considerable agreement as to the type of learning outcomes valued and the different types of classroom activity that these outcomes might imply. The following non-hierarchical list was found helpful:

(R.Hyde)




• •



• • • •



• • • •


• • •


• • • •




Comparing ideal values with values in practice When participants were asked to compare their “vision for an ideal mathematics curriculum” with the values that are implied by the “curriculum that is currently implemented in most schools and other settings”, a consistent pattern emerged. Participants recorded on a four-point scale how frequently mathematics lessons should include each learning outcome and also how frequently mathematics lessons, in their view, actually do reflect each learning outcome (1 = hardly ever, 4 = almost every lesson). The aggregated responses of 133 initial conference and colloquia participants are shown in the graph below. The results show that participants consider that fluency is overvalued in the current curriculum, while the remaining values are undervalued. Perhaps surprisingly, this pattern of response was repeated consistently across responses from each phase of education (primary, secondary and tertiary) and across all regions. We note again, however, that due to the small and self-selecting nature of the sample involved (particularly with regard to primary teachers) these results may not reflect those of the profession as a whole. All respondents: Ideal and Current Outcomes – Weighted Average where high = most important

Ideal Curriculum Current Curriculum 4

N = 133

Source: (Swan, 2006 p.42)

3.5

3

Weighted Average


2.5

2

1.5

1

0.5

…a way of making meanings, of viewing and describing the world. (P.Lacey, P.Drake)

10

0 Fluency in recalling facts and performing skills

Interpretations for concepts and representations


11


Appreciation of the power of mathematics

Illustrating different types of learning outcome Worthwhile learning tasks are likely to reflect more than one of the purposes set out above, but there may be a tension if teaching attempts to serve incompatible purposes at the same time. For example, if learners are asked to ‘investigate’ an open situation, while the teacher wants them to ‘discover’ a culturally valued result (such as Pythagoras’ theorem), an incompatibility arises between the divergence of the learners’ explorations and the convergent purposes of the teacher. Bearing this in mind, participants at the initial conference were invited to describe inspiring examples of lessons they had seen and identify the purposes being served. Below, we offer five of their contributions, and identify each one with a different purpose. Some of the contributions relate to specific phases, though some of the ideas may be adapted for use in other phases.

Example 1. Fluency in recalling facts and performing skills (A. Watson)

Example 2. Interpretations for concepts and representations (P. Lacey)

Learners constructing examples introduces a sense of the structure, characteristics and properties of new objects. This example, from a secondary classroom, also includes practice and fluency. The ‘big’ idea of conjugation is being introduced building on successful past tool use of the grid. There is a sense of ownership, mystery, intrigue. Also it feels do-able, but the difficulty adds to the motivation. Also, you don’t have to tell learners everything – or show them.

Extrapolating new knowledge from old is an engine for exploring and charting the territory of mathematics. In this exercise learners are expected to apply their understanding of fundamental concepts and principles (inverse, equivalence etc) in order to ‘map’ and extend their knowledge.

What was the mathematical task? Having been just introduced to the idea that there are numbers of the form a + √b, and reminded of grid multiplication, learners were asked to find two such numbers which, when multiplied, you ‘lose’ the root part.

How could you tell that the task achieved the intended purposes? I saw the written work. Some aspects of purpose achieved by all: (a+√b)(a-√b) = integer found by some. Is this example available to see/read about? It will be (special issue of Educational Studies in Mathematics 2008) Can you say why you chose this example? What criteria were in your mind? To illustrate use of learner-generated examples and to promote shifts between empirical and structural reasoning. 12

How could you tell that the task achieved the intended purposes? I think the activity actually altered the views of some of the learners on what mathematics actually is. Certainly challenged the ‘quantum’ view of mathematics as isolated facts.

What was the mathematical task? What else do you know (and why) if you know that 5 + 3 = 8? The statement was written in the middle of a board visible to a whole class. Initially whole class responses. For example: 500 + 300 = 800. This was written on the board with an arrow to it from the original statement. A ‘mindmap’ was generated with answers to the ‘why’ bit of the question determining the connections. This lead on to learners working in groups/pairs to create a ‘composite’ mind map. Discussing the ‘why’ proved productive.

Learners worked on their own or in pairs. All rough working was kept and it was clear that most learners shifted from some sort of ‘testing’ various integers’ approach to a ‘structural’ approach, e.g.: trying (2+√3)(3+√2) or (2+√2)(2+√2) etc. Many started by using calculator, some abandoned this as they focused on structure. What learning culture was created? How was this achieved? Learners trusted the teacher because of having done similar tasks in the past. They were allowed to choose starting numbers and methods of working, except for grid multiplication. There was lots of talk. Some ‘gossiped’ the method (distributed knowledge!). Teacher asked ‘why choose this, why choose that?’ and reminded them of purpose. Some of the learners carried on after class together or separately and the teacher was interested in their work!

What learning culture was created? How was this achieved? Learners were in control. There was no perceived limit. There was genuine sharing of personal understandings.

Is this example available to see/read about? Reported in ‘Mathematics Teaching’ 187 June 2004 as a special conference insert, after being included in an ATM annual conference presentation. Can you say why you chose this example? What criteria were in your mind? Simple and accessible start – almost trivial; but deep in its engagement. Explicit and shared discussions on personal maps of understanding have a sense of deep learning.

150 ÷ 30 = 5 15 ÷ 3 = 5 3 x 5 = 15 2 x 5 = 10

5 + 6 = 11

90 – 50 = 40

50 + 40 = 90 50 + 30 = 80

9–5=4

5 + 5 = 10

80 – 50 = 30

5+3=8

5+4=9

190 – 150 = 40

8–3=5

19 – 15 = 4 8–5=3

800 – 300 = 500

9–5=4

8–4=4 823 – 323 = 500 80 – 50 = 30

180 – 150 = 30

13

Illustrating different types of learning outcome Worthwhile learning tasks are likely to reflect more than one of the purposes set out above, but there may be a tension if teaching attempts to serve incompatible purposes at the same time. For example, if learners are asked to ‘investigate’ an open situation, while the teacher wants them to ‘discover’ a culturally valued result (such as Pythagoras’ theorem), an incompatibility arises between the divergence of the learners’ explorations and the convergent purposes of the teacher. Bearing this in mind, participants at the initial conference were invited to describe inspiring examples of lessons they had seen and identify the purposes being served. Below, we offer five of their contributions, and identify each one with a different purpose. Some of the contributions relate to specific phases, though some of the ideas may be adapted for use in other phases.

Example 1. Fluency in recalling facts and performing skills (A. Watson)

Example 2. Interpretations for concepts and representations (P. Lacey)

Learners constructing examples introduces a sense of the structure, characteristics and properties of new objects. This example, from a secondary classroom, also includes practice and fluency. The ‘big’ idea of conjugation is being introduced building on successful past tool use of the grid. There is a sense of ownership, mystery, intrigue. Also it feels do-able, but the difficulty adds to the motivation. Also, you don’t have to tell learners everything – or show them.

Extrapolating new knowledge from old is an engine for exploring and charting the territory of mathematics. In this exercise learners are expected to apply their understanding of fundamental concepts and principles (inverse, equivalence etc) in order to ‘map’ and extend their knowledge.

What was the mathematical task? Having been just introduced to the idea that there are numbers of the form a + √b, and reminded of grid multiplication, learners were asked to find two such numbers which, when multiplied, you ‘lose’ the root part.

How could you tell that the task achieved the intended purposes? I saw the written work. Some aspects of purpose achieved by all: (a+√b)(a-√b) = integer found by some. Is this example available to see/read about? It will be (special issue of Educational Studies in Mathematics 2008) Can you say why you chose this example? What criteria were in your mind? To illustrate use of learner-generated examples and to promote shifts between empirical and structural reasoning. 12

How could you tell that the task achieved the intended purposes? I think the activity actually altered the views of some of the learners on what mathematics actually is. Certainly challenged the ‘quantum’ view of mathematics as isolated facts.

What was the mathematical task? What else do you know (and why) if you know that 5 + 3 = 8? The statement was written in the middle of a board visible to a whole class. Initially whole class responses. For example: 500 + 300 = 800. This was written on the board with an arrow to it from the original statement. A ‘mindmap’ was generated with answers to the ‘why’ bit of the question determining the connections. This lead on to learners working in groups/pairs to create a ‘composite’ mind map. Discussing the ‘why’ proved productive.

Learners worked on their own or in pairs. All rough working was kept and it was clear that most learners shifted from some sort of ‘testing’ various integers’ approach to a ‘structural’ approach, e.g.: trying (2+√3)(3+√2) or (2+√2)(2+√2) etc. Many started by using calculator, some abandoned this as they focused on structure. What learning culture was created? How was this achieved? Learners trusted the teacher because of having done similar tasks in the past. They were allowed to choose starting numbers and methods of working, except for grid multiplication. There was lots of talk. Some ‘gossiped’ the method (distributed knowledge!). Teacher asked ‘why choose this, why choose that?’ and reminded them of purpose. Some of the learners carried on after class together or separately and the teacher was interested in their work!

What learning culture was created? How was this achieved? Learners were in control. There was no perceived limit. There was genuine sharing of personal understandings.

Is this example available to see/read about? Reported in ‘Mathematics Teaching’ 187 June 2004 as a special conference insert, after being included in an ATM annual conference presentation. Can you say why you chose this example? What criteria were in your mind? Simple and accessible start – almost trivial; but deep in its engagement. Explicit and shared discussions on personal maps of understanding have a sense of deep learning.

150 ÷ 30 = 5 15 ÷ 3 = 5 3 x 5 = 15 2 x 5 = 10

5 + 6 = 11

90 – 50 = 40

50 + 40 = 90 50 + 30 = 80

9–5=4

5 + 5 = 10

80 – 50 = 30

5+3=8

5+4=9

190 – 150 = 40

8–3=5

19 – 15 = 4 8–5=3

800 – 300 = 500

9–5=4

8–4=4 823 – 323 = 500 80 – 50 = 30

180 – 150 = 30

13

Example 3. Strategies for investigation and problem solving (J. Edwards)

Example 4. Awareness of the nature and values of the educational system (M. Swan)

How could you tell that the task achieved the intended purposes? Do you have any evidence? We didn’t collect evidence on the mathematics learned, but the reflective thought that went into the review and design of the teaching sessions was considerable. Pupils began to show evidence of appreciating that teaching is not just about ‘telling’. When, for example, the girl above was asked about the experience, she responded as follows: (Int = interviewer):

What was the mathematical task? This task was designed to help learners become aware of what it feels like to teach something, and in so doing to review and reorganise their own understanding of a topic.

FINAL SCORE The final score in a hockey game was 2 – 1. List the possible half-time scores. How many possible scores are there? Try this for other scores.

From: Graded Assessment in Mathematics (GAIM) Macmillan Education 1988 The GCSE class was set 10 of 10, a group of 3 girls. In the previous lesson, learners had generated enough data to generalise and after considerable debate, they arrived at the number of possible half-time scores being (x+1)(y+1) where x and y are the final scores. When challenged to explain why this expression ‘worked’, they argued, challenged, justified amongst themselves for half an hour. Eventually, they were able to describe why they needed to add one to each score. The value of this is was in: (a) being able to comfortably challenge each other; (b) achieving ‘hard’ maths (for them); (c) being aware that mathematics can describe situations d) being able to interpret a situation mathematically.

How could you tell that the task achieved the intended purposes? Being able to communicate the outcomes and process both verbally and in writing. Pleasure expressed by the pupils about their achievement. (Despite being in set 10 of 10, this ‘diet’ of mathematics learning resulted in GCSE grades D,E,E for this group of girls.) Is this example available to see/read about? Graded Assessment in Mathematics (GAIM) Macmillan Education 1988 p68. Can you say why you chose this example? What criteria were in your mind? Level of enjoyment for pupils; Level of challenge; Potential for awareness that mathematics can describe ‘real’ situations; Potential to develop confidence – ‘doing quadratics’ is for ‘good’ people.

What learning culture was created? How was this achieved? A background of feminist epistemology. The 10 school ‘classroom rules’ were abandoned in favour of 3 mathematics classroom rules:

• • •

everyone does mathematics everyone does mathematics in a way that enables others to do mathematics everyone shares their mathematics (collaborates).

What was the mathematical task? Two parallel year 7 classes were taught different topics by different teachers. One class was taught “Area” and the other was taught "Bearings". At the end of this period, the teachers told their learners that they would now be given two lessons to prepare to teach the topic they had just learned to members of the other class.

Int: Girl: Int: Girl: Int: Girl:

Learners were given some advice on how to go about this process. They were asked to decide exactly what the big ideas in the topic were, the order these should be presented in, to identify difficult ideas and prepare resources help to explain them (e.g. a worksheet). They were also asked to think how they would find out if their teaching had been successful. When both classes were ready, half of the pupils moved to the other classroom and each one sat alongside a pupil from the other group and taught them the topic. In a second lesson, roles were reversed.

Is this example available to see/read about? This is referred to in (Bell, A; Crust, R; Shannon, A; Swan, M; “Awareness of Learning, Reflection and Transfer in School Mathematics“, ESRC Project report: R000-23-2329, Shell Centre for Mathematical Education, University of Nottingham) Can you say why you chose this example? What criteria were in your mind? To illustrate how changing classroom roles can begin to develop pupils’ awareness of the nature of teaching and learning itself. This can be done alongside the development of mathematical skills and concepts.

What learning culture was created? How was this achieved? Learners appeared nervous but very motivated by this challenge. They took a great deal of trouble in creating attractive and informative worksheets and some even had them typed out and checked beforehand. During the teaching phase, learners adopted teacher roles with some enthusiasm. One girl (teaching two boys due to uneven class sizes), adopted a very confident style, making statements like: “How do you find the area from these two numbers? You can draw centimetre squares if you want.” “ Talk then or you won’t learn anything!” “Pretend you are the teacher, how would you explain it to me?” “Put your hands up if you don't know.”

Small groups were based on friendship, there was an ethos of collaboration, feminist principles of connectedness. A shared understanding that a mathematics classroom is where thinking takes place.

The boys, it must be said, couldn’t wait for their turn to teach! This positive culture was achieved because these teachers had high expectations for their classes and wanted their learners to take active roles in lessons.

14

What was hard about being a teacher? After teaching for some time, I find they don't understand. How did the experience help you? It gave me more confidence. Would you like to try teaching again? Yes, so what I've learned now I must try on other pupils so I get better at teaching. I can’t understand how to teach well.

15

Example 3. Strategies for investigation and problem solving (J. Edwards)

Example 4. Awareness of the nature and values of the educational system (M. Swan)

How could you tell that the task achieved the intended purposes? Do you have any evidence? We didn’t collect evidence on the mathematics learned, but the reflective thought that went into the review and design of the teaching sessions was considerable. Pupils began to show evidence of appreciating that teaching is not just about ‘telling’. When, for example, the girl above was asked about the experience, she responded as follows: (Int = interviewer):

What was the mathematical task? This task was designed to help learners become aware of what it feels like to teach something, and in so doing to review and reorganise their own understanding of a topic.

FINAL SCORE The final score in a hockey game was 2 – 1. List the possible half-time scores. How many possible scores are there? Try this for other scores.

From: Graded Assessment in Mathematics (GAIM) Macmillan Education 1988 The GCSE class was set 10 of 10, a group of 3 girls. In the previous lesson, learners had generated enough data to generalise and after considerable debate, they arrived at the number of possible half-time scores being (x+1)(y+1) where x and y are the final scores. When challenged to explain why this expression ‘worked’, they argued, challenged, justified amongst themselves for half an hour. Eventually, they were able to describe why they needed to add one to each score. The value of this is was in: (a) being able to comfortably challenge each other; (b) achieving ‘hard’ maths (for them); (c) being aware that mathematics can describe situations d) being able to interpret a situation mathematically.

How could you tell that the task achieved the intended purposes? Being able to communicate the outcomes and process both verbally and in writing. Pleasure expressed by the pupils about their achievement. (Despite being in set 10 of 10, this ‘diet’ of mathematics learning resulted in GCSE grades D,E,E for this group of girls.) Is this example available to see/read about? Graded Assessment in Mathematics (GAIM) Macmillan Education 1988 p68. Can you say why you chose this example? What criteria were in your mind? Level of enjoyment for pupils; Level of challenge; Potential for awareness that mathematics can describe ‘real’ situations; Potential to develop confidence – ‘doing quadratics’ is for ‘good’ people.

What learning culture was created? How was this achieved? A background of feminist epistemology. The 10 school ‘classroom rules’ were abandoned in favour of 3 mathematics classroom rules:

• • •

everyone does mathematics everyone does mathematics in a way that enables others to do mathematics everyone shares their mathematics (collaborates).

What was the mathematical task? Two parallel year 7 classes were taught different topics by different teachers. One class was taught “Area” and the other was taught "Bearings". At the end of this period, the teachers told their learners that they would now be given two lessons to prepare to teach the topic they had just learned to members of the other class.

Int: Girl: Int: Girl: Int: Girl:

Learners were given some advice on how to go about this process. They were asked to decide exactly what the big ideas in the topic were, the order these should be presented in, to identify difficult ideas and prepare resources help to explain them (e.g. a worksheet). They were also asked to think how they would find out if their teaching had been successful. When both classes were ready, half of the pupils moved to the other classroom and each one sat alongside a pupil from the other group and taught them the topic. In a second lesson, roles were reversed.

Is this example available to see/read about? This is referred to in (Bell, A; Crust, R; Shannon, A; Swan, M; “Awareness of Learning, Reflection and Transfer in School Mathematics“, ESRC Project report: R000-23-2329, Shell Centre for Mathematical Education, University of Nottingham) Can you say why you chose this example? What criteria were in your mind? To illustrate how changing classroom roles can begin to develop pupils’ awareness of the nature of teaching and learning itself. This can be done alongside the development of mathematical skills and concepts.

What learning culture was created? How was this achieved? Learners appeared nervous but very motivated by this challenge. They took a great deal of trouble in creating attractive and informative worksheets and some even had them typed out and checked beforehand. During the teaching phase, learners adopted teacher roles with some enthusiasm. One girl (teaching two boys due to uneven class sizes), adopted a very confident style, making statements like: “How do you find the area from these two numbers? You can draw centimetre squares if you want.” “ Talk then or you won’t learn anything!” “Pretend you are the teacher, how would you explain it to me?” “Put your hands up if you don't know.”

Small groups were based on friendship, there was an ethos of collaboration, feminist principles of connectedness. A shared understanding that a mathematics classroom is where thinking takes place.

The boys, it must be said, couldn’t wait for their turn to teach! This positive culture was achieved because these teachers had high expectations for their classes and wanted their learners to take active roles in lessons.

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What was hard about being a teacher? After teaching for some time, I find they don't understand. How did the experience help you? It gave me more confidence. Would you like to try teaching again? Yes, so what I've learned now I must try on other pupils so I get better at teaching. I can’t understand how to teach well.

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Example 5. Appreciation of the power of mathematics in society (D. Pratt)

What learning culture was created? How was this achieved? No one task generates a culture. Indeed, a task can create discipline problems if the expectations of behaviour in the task clash with the prevailing culture. In this task, we were aiming for a culture in which mathematics is used to explore an interesting problem, resulting in the learning of new mathematics. Learners needed to work in small groups and work together, sharing roles. The mathematics was seen as powerful.

What was the mathematical task? What makes the best design for a paper ‘helicopter’? Make a spinner and try it out. Compare 2 spinners. Why did you like that one? What variables affect the flight (usually time of flight) In groups, explore the variable (e.g. length of wings) comparing it to the dependent variable (time). Perhaps use Active Graphing: (a) Enter a few ordinate points into a spreadsheet (b) Generate a scatter-graph (c) Decide what to do next in the experiment (d) Go to (a).

• • • • • • •

How could you tell that the task achieved the intended purposes? Learners: used graphs to make decision about where to take the experiment used data analysis methods to increase accuracy of experiment worked productively together could describe relationships between the independent and dependent variables.

• • • •

Is this example available to see/read about? Ainley, J., Pratt, D. and Hansen, A. Connecting engagement and focus in pedagogic task design; British Educational Research Journal. 32.1, 23-38. 2006.

Organisation: Norton College

Age/ability range Year 7 class mixed boys and girls. The students were studying probability concepts both theoretically and experimentally. How was the session/task introduced? I had a slide of the old-fashioned ‘pinball’ type toy where a ball can be dropped in from the top and the ball makes its way to bins at the bottom. Several students recognised this type of equipment and we discussed what might happen to balls as they were dropped into the pinball machine. The discussion was good with various ideas about how the bins might fill up; an equal distribution of balls in the slots was a common thought.

How was the session/task sustained? After generating interest in the outcomes, I showed them a virtual pinball simulator available as a Java type animation. (Web address of applet: http://www.jcu.edu/math/isep/Quincunx/Quincunx.html ) We experimented with the virtual pinball as a whole class activity, dropping balls of various sample sizes and with different numbers of bins at the bottom of the machine. A helpful visual feature of the applet was the generation of a frequency diagram revealing the way in which the balls fell. The distribution of balls wasn’t really what the class had generally expected. Claims of bad luck and fluke results were a general explanation for the distributions generated, so we re-ran the pinball simulator several times.

Ainley, J., Pratt, D., and Nardi, E; Normalising: children’s activity to construct meanings for trend; Educational Studies in Mathematics, 45, 131-146. 2001.

Amongst the class there was much debate about why the results were not as expected. Some students began to venture explanations for the distribution and started to talk about the ‘routes’ that led to the bins at the bottom of the pinball machine.

Ainley, J. Super 'copters Micromath Autumn 1991.

How was the session/task concluded? Students suggested we reduce the simulator to a small number of bins and we then attempted to analyse the ways in which the balls could reach the bins. Within a short space of time, explanations for the results yielded by the simulator were produced. We verified our thoughts with some attempts to predict the number of balls our theoretical models were predicting.

Can you say why you chose this example? What criteria were in your mind? Engaging learners (purpose); Engagement leading to focus (utility); Problem solving; Awareness of the power of mathematics (graphing as an analytical tool; average for smoothing errors). I have used the example many times and at many levels. It seems to provide an opportunity for thinking of graphs as analytical tools, rather than as presentational tools. Chosen out of frustration at narrow use of graphing in schools.

At the beginning of each colloquium, participants were asked to describe a lesson that they had found memorable and inspiring. In total, we received 57 lesson accounts, which have been checked and validated and are now available on the NCETM portal www.ncetm.org.uk Each is categorised in the following ways:

Name: Dan Curran

What were the critical moments? When the students started to “list” the possible routes to the slots, a connection to the structure of the pinball machine became apparent. They generated the routes themselves – albeit this was often attempted in a haphazard and unstructured way. A second feature was the notion of experimental probability compared with the theoretical model. Lots of good discussion about ideas of luck and chance influencing deviations from our expectations. The ability to verify or discount ideas quickly made the debate manageable.

3. By the nature of the learner group. A sample lesson account is given below. The teacher has classified this as developing “strategies for investigation and problem solving”; and as illustrating “encourages reasoning rather than ‘answer getting’” and “creates connections between topics” for 11–16.

1. By the values that underpin the purpose of it.

What mathematics was learnt? Apart from the theoretical and experimental probability ideas, there were opportunities for discussion of how the results were being represented graphically. An unexpected bonus was that the class was able to consider very large samples. How was that mathematics learnt? Mainly discussion and recording the ‘routes’ through the pinball machine. Resources The lesson was displayed through the data projector and interactive whiteboard.

2. By the principles being demonstrated.

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Example 5. Appreciation of the power of mathematics in society (D. Pratt)

What learning culture was created? How was this achieved? No one task generates a culture. Indeed, a task can create discipline problems if the expectations of behaviour in the task clash with the prevailing culture. In this task, we were aiming for a culture in which mathematics is used to explore an interesting problem, resulting in the learning of new mathematics. Learners needed to work in small groups and work together, sharing roles. The mathematics was seen as powerful.

What was the mathematical task? What makes the best design for a paper ‘helicopter’? Make a spinner and try it out. Compare 2 spinners. Why did you like that one? What variables affect the flight (usually time of flight) In groups, explore the variable (e.g. length of wings) comparing it to the dependent variable (time). Perhaps use Active Graphing: (a) Enter a few ordinate points into a spreadsheet (b) Generate a scatter-graph (c) Decide what to do next in the experiment (d) Go to (a).

• • • • • • •

How could you tell that the task achieved the intended purposes? Learners: used graphs to make decision about where to take the experiment used data analysis methods to increase accuracy of experiment worked productively together could describe relationships between the independent and dependent variables.

• • • •

Is this example available to see/read about? Ainley, J., Pratt, D. and Hansen, A. Connecting engagement and focus in pedagogic task design; British Educational Research Journal. 32.1, 23-38. 2006.

Organisation: Norton College

Age/ability range Year 7 class mixed boys and girls. The students were studying probability concepts both theoretically and experimentally. How was the session/task introduced? I had a slide of the old-fashioned ‘pinball’ type toy where a ball can be dropped in from the top and the ball makes its way to bins at the bottom. Several students recognised this type of equipment and we discussed what might happen to balls as they were dropped into the pinball machine. The discussion was good with various ideas about how the bins might fill up; an equal distribution of balls in the slots was a common thought.

How was the session/task sustained? After generating interest in the outcomes, I showed them a virtual pinball simulator available as a Java type animation. (Web address of applet: http://www.jcu.edu/math/isep/Quincunx/Quincunx.html ) We experimented with the virtual pinball as a whole class activity, dropping balls of various sample sizes and with different numbers of bins at the bottom of the machine. A helpful visual feature of the applet was the generation of a frequency diagram revealing the way in which the balls fell. The distribution of balls wasn’t really what the class had generally expected. Claims of bad luck and fluke results were a general explanation for the distributions generated, so we re-ran the pinball simulator several times.

Ainley, J., Pratt, D., and Nardi, E; Normalising: children’s activity to construct meanings for trend; Educational Studies in Mathematics, 45, 131-146. 2001.

Amongst the class there was much debate about why the results were not as expected. Some students began to venture explanations for the distribution and started to talk about the ‘routes’ that led to the bins at the bottom of the pinball machine.

Ainley, J. Super 'copters Micromath Autumn 1991.

How was the session/task concluded? Students suggested we reduce the simulator to a small number of bins and we then attempted to analyse the ways in which the balls could reach the bins. Within a short space of time, explanations for the results yielded by the simulator were produced. We verified our thoughts with some attempts to predict the number of balls our theoretical models were predicting.

Can you say why you chose this example? What criteria were in your mind? Engaging learners (purpose); Engagement leading to focus (utility); Problem solving; Awareness of the power of mathematics (graphing as an analytical tool; average for smoothing errors). I have used the example many times and at many levels. It seems to provide an opportunity for thinking of graphs as analytical tools, rather than as presentational tools. Chosen out of frustration at narrow use of graphing in schools.

At the beginning of each colloquium, participants were asked to describe a lesson that they had found memorable and inspiring. In total, we received 57 lesson accounts, which have been checked and validated and are now available on the NCETM portal www.ncetm.org.uk Each is categorised in the following ways:

Name: Dan Curran

What were the critical moments? When the students started to “list” the possible routes to the slots, a connection to the structure of the pinball machine became apparent. They generated the routes themselves – albeit this was often attempted in a haphazard and unstructured way. A second feature was the notion of experimental probability compared with the theoretical model. Lots of good discussion about ideas of luck and chance influencing deviations from our expectations. The ability to verify or discount ideas quickly made the debate manageable.

3. By the nature of the learner group. A sample lesson account is given below. The teacher has classified this as developing “strategies for investigation and problem solving”; and as illustrating “encourages reasoning rather than ‘answer getting’” and “creates connections between topics” for 11–16.

1. By the values that underpin the purpose of it.

What mathematics was learnt? Apart from the theoretical and experimental probability ideas, there were opportunities for discussion of how the results were being represented graphically. An unexpected bonus was that the class was able to consider very large samples. How was that mathematics learnt? Mainly discussion and recording the ‘routes’ through the pinball machine. Resources The lesson was displayed through the data projector and interactive whiteboard.

2. By the principles being demonstrated.

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learners to notice ‘sameness’ and ‘difference’, examine alternative representations and share and discuss interpretations. In order to develop problem solving strategies, teachers need to allow learners to compare the outcomes of applying different strategies to an unstructured problem and examine the advantages and disadvantages of each. Different goals clearly make different pedagogical demands on a teacher.

Principles that guide teaching In articulating their values, participants began to clarify the learning goals and classroom activities they seek to achieve. They now attempted to identify research-based principles that could be used to inform the way they teach towards these goals. Different goals require different types of learning activity and an appropriate choice of teaching strategy. For example, in order to achieve fluency in the recall of facts or the performance of skills, it is appropriate to emphasise personal practice. In order to develop conceptual understanding however, the teacher must encourage

Principles for effective teaching therefore depend on the purposes they seek to address. The recent “Improving Learning in Mathematics” materials, focused on the conceptual development and sought to embody the following list of research-based principles (DfES, 2005).

Teaching is more effective when it…

At the colloquia, 64% (46/72) of participants totally agreed with the principles presented, 32% mainly agreed, expressing reservations, while 4% expressed particular concerns. No participants stated that they disagreed with the principles.

Participants proposed a number of amendments to the principles, around using whole class interactive teaching; creating connections within mathematics and in the real world; and using all resources creatively . Participants proposed some additions to the principles around confronting difficulties; developing mathematical language; and recognising learning.

Of those who expressed some reservation or concern, 35% (9/26) related to the use of technology – it should be used as a tool and is not always necessary. Colleagues working outside traditional educational settings pointed out that technology is not always available. 23% cited concerns about the confusion that may be caused when exposing and discussing common misconceptions (R.Archer, J.Bean, M.Dovey).

Thus, we have retained all the original principles, with slight rewording to 2, 4, 7, 8:


1. Builds on the knowledge learners already have

This means developing formative assessment techniques and adapting our teaching to accommodate individual learning needs.



2. Exposes and discusses common misconceptions

Learning activities should exposing current thinking, create ‘tensions’ by confronting learners with inconsistencies, and allow opportunities for resolution through discussion.

2. Exposes and discusses common misconceptions and other surprising phenomena

Learning activities should expose current thinking, create ‘tensions’ by confronting learners with inconsistencies and surprises, and allow opportunities for resolution through discussion.

3. Uses higher-order questions

Questioning is more effective when it promotes explanation, application and synthesis rather than mere recall.



Activities are more effective when they encourage critical, constructive discussion, rather than argumentation or uncritical acceptance. Shared goals and group accountability are important.

4. Makes appropriate use of whole class interactive teaching, individual work and cooperative small group work

Collaborative group work is more effective after learners have been given an opportunity for individual reflection.

4. Uses cooperative small group work

5. Encourages reasoning rather than ‘answer getting’

Often, learners are more concerned with what they have ‘done’ than with what they have learned. It is better to aim for depth than for superficial ‘coverage’.

6. Uses rich, collaborative tasks

The tasks used should be accessible, extendable, encourage decision-making, promote discussion, encourage creativity, encourage ‘what if’ and ‘what if not’ questions.


Learners often find it difficult to generalise and transfer their learning to other topics and contexts. Related concepts (such as division, fraction and ratio) remain unconnected. Effective teachers build bridges between ideas.

7. Creates connections between topics

Computers, interactive whiteboards and graphics calculators offer learners opportunities to explore rich ‘microworlds’, and to represent, analyse and communicate mathematics with interactive, dynamic tools.

8. Uses technology in appropriate ways




The tasks used should be accessible, extendable, encourage decision-making, promote discussion, encourage creativity, encourage ‘what if’ and what if not’ questions.

7. Creates connections between topics both within and beyond mathematics and with the real world


8. Uses resources, including technology, in creative and appropriate ways

ICT offers new ways to engage with mathematics. At its best it is dynamic and visual: relationships become more tangible. ICT can provide feedback on actions and enhance interactivity and learner autonomy. Through its connectivity, ICT offers the means to access and share resources and – even more powerfully – the means by which learners can share their ideas within and across classrooms.

Teaching is more effective when it develops awareness of the ‘bigger picture’. (M.Horner, R.Hyde, J.Garvey, Jones, P.Lacey, Taylor, G.Wake)

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learners to notice ‘sameness’ and ‘difference’, examine alternative representations and share and discuss interpretations. In order to develop problem solving strategies, teachers need to allow learners to compare the outcomes of applying different strategies to an unstructured problem and examine the advantages and disadvantages of each. Different goals clearly make different pedagogical demands on a teacher.

Principles that guide teaching In articulating their values, participants began to clarify the learning goals and classroom activities they seek to achieve. They now attempted to identify research-based principles that could be used to inform the way they teach towards these goals. Different goals require different types of learning activity and an appropriate choice of teaching strategy. For example, in order to achieve fluency in the recall of facts or the performance of skills, it is appropriate to emphasise personal practice. In order to develop conceptual understanding however, the teacher must encourage

Principles for effective teaching therefore depend on the purposes they seek to address. The recent “Improving Learning in Mathematics” materials, focused on the conceptual development and sought to embody the following list of research-based principles (DfES, 2005).


At the colloquia, 64% (46/72) of participants totally agreed with the principles presented, 32% mainly agreed, expressing reservations, while 4% expressed particular concerns. No participants stated that they disagreed with the principles.

Participants proposed a number of amendments to the principles, around using whole class interactive teaching; creating connections within mathematics and in the real world; and using all resources creatively . Participants proposed some additions to the principles around confronting difficulties; developing mathematical language; and recognising learning.

Of those who expressed some reservation or concern, 35% (9/26) related to the use of technology – it should be used as a tool and is not always necessary. Colleagues working outside traditional educational settings pointed out that technology is not always available. 23% cited concerns about the confusion that may be caused when exposing and discussing common misconceptions (R.Archer, J.Bean, M.Dovey).

Thus, we have retained all the original principles, with slight rewording to 2, 4, 7, 8:






2. Exposes and discusses common misconceptions

Learning activities should exposing current thinking, create ‘tensions’ by confronting learners with inconsistencies, and allow opportunities for resolution through discussion.

2. Exposes and discusses common misconceptions and other surprising phenomena

Learning activities should expose current thinking, create ‘tensions’ by confronting learners with inconsistencies and surprises, and allow opportunities for resolution through discussion.






4. Makes appropriate use of whole class interactive teaching, individual work and cooperative small group work

Collaborative group work is more effective after learners have been given an opportunity for individual reflection.

4. Uses cooperative small group work




The tasks used should be accessible, extendable, encourage decision-making, promote discussion, encourage creativity, encourage ‘what if’ and ‘what if not’ questions.



7. Creates connections between topics

Computers, interactive whiteboards and graphics calculators offer learners opportunities to explore rich ‘microworlds’, and to represent, analyse and communicate mathematics with interactive, dynamic tools.

8. Uses technology in appropriate ways




The tasks used should be accessible, extendable, encourage decision-making, promote discussion, encourage creativity, encourage ‘what if’ and what if not’ questions.

7. Creates connections between topics both within and beyond mathematics and with the real world


8. Uses resources, including technology, in creative and appropriate ways

ICT offers new ways to engage with mathematics. At its best it is dynamic and visual: relationships become more tangible. ICT can provide feedback on actions and enhance interactivity and learner autonomy. Through its connectivity, ICT offers the means to access and share resources and – even more powerfully – the means by which learners can share their ideas within and across classrooms.

Teaching is more effective when it develops awareness of the ‘bigger picture’. (M.Horner, R.Hyde, J.Garvey, Jones, P.Lacey, Taylor, G.Wake)

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We have also added the following principles, which we believe can also be substantiated from research: Teaching is more effective when it… Effective teaching challenges learners and has high expectations of them. It does not seek to ‘smooth the path’ but creates realistic obstacles to be overcome. Confidence, persistence and learning are not attained through repeating successes, but by struggling with difficulties.

9. Confronts difficulties rather than seeks to avoid or pre-empt them

Mathematics is a language that enables us to describe and model situations, think logically, frame and sustain arguments and communicate ideas with precision. Learners do not know mathematics until they can ‘speak’ it. Effective teaching therefore focuses on the communicative aspects of mathematics by developing oral and written mathematical language.

10. Develops mathematical language through communicative activities

What is to be learned cannot always be stated prior to the learning experience. After a learning event, however, it is important to reflect on the learning that has taken place, making this as explicit and memorable as possible. Effective teachers will also reflect on the ways in which learning has taken place, so that learners develop their own capacity to learn.

11. Recognises both what has been learned and also how it has been learned

Teaching is more effective when it develops mathematical language through communicative activities.



We invited participants to describe any principles that they believe are commonly held, but which are positively unhelpful to teachers.

At the initial conference and the subsequent colloquia, participants were invited to respond to the two open questions:

Participants generated the following list of what they considered to be unhelpful advice:

• • • • • • • • • • • • •

What are the major obstacles to progress? How do these obstacles function? These questions unlocked some strong feelings among participants. The responses are categorised into five overlapping themes, below. It must be emphasised that these themes were not put to participants, they emerged from the responses.

Learn how to do it first – understanding will come later. Repetition will improve understanding. There is a ‘best way’ to teach, an ‘optimal sequence’ for learning, a ‘right way’ to solve each problem. Explain clearly how to do the problem before you give it to your class. Learning must be preceded by instruction. Tell the class your lesson objectives at the beginning of each lesson. Make sure your lesson has three parts. Plan the plenary discussion very carefully and stick to your plan. Cover the syllabus, even if this means hurrying through some parts. Keep emphasising presentation and neatness. Knowing the answer is important. Keep learners busy. Learners go off-task if they talk. Don’t confuse learners by showing them incorrect methods. Use technology whenever you can.

1. Society’s attitude towards mathematics Participants felt that society and its media have a limited, narrow view of mathematics and still promote it as a subject for ‘geeks’. Mathematical incompetence continues to remain socially acceptable. Some also commented on the climate of distrust of teachers and the excessively detailed accountability that permeates and distorts the teaching of mathematics. 2. Teachers’ subject and pedagogical subject knowledge Participants felt that many teachers lack confidence in the subject and an awareness and understanding of appropriate resources (including ICT). They also lack time for continuing professional development. Participants commented that many teachers ‘lack confidence’ in mathematics, ‘can’t see the bigger, interconnected picture’ and ‘can only see one learning pathway through the curriculum’. When unexpected insights, answers or misconceptions arise, for example, many teachers don’t have sufficient subject knowledge to depart from their predetermined plan. This is also evidenced by the overdependence on pre-packaged schemes and textbooks.

Most of these ’unhelpful principles’ are well-intentioned but they become unhelpful when their continued emphasis encourages superficial activity rather than deeper learning.

(B.Vertes)

Mathematics teachers have to overcome society’s attitudes: – that mathematics is “geeky” rather than “cool” – that it is acceptable not to be good at mathematics.

One participant asserted that a common view of mathematics shared by teachers and pupils is that it is a ‘dead weight of predetermined knowledge to be absorbed’, rather than an organic system to explore collaboratively. In attempting to deal with learner disaffection, participants felt that many reward systems encourage extrinsic motivation, rather than an intrinsic satisfaction in the subject itself. They also felt that there is a developing practice of over-praising or praising too soon and teachers being trained to control, not empower.

(P.Williams, H.Pomroy, E.Lawrenson, L.Bell, M.Thomas, M.Elliot)

Compounding this, it was felt that there is a lack of subject-specific CPD within schools and colleges, with opportunities for research and reflection.

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We have also added the following principles, which we believe can also be substantiated from research: Teaching is more effective when it… Effective teaching challenges learners and has high expectations of them. It does not seek to ‘smooth the path’ but creates realistic obstacles to be overcome. Confidence, persistence and learning are not attained through repeating successes, but by struggling with difficulties.

9. Confronts difficulties rather than seeks to avoid or pre-empt them

Mathematics is a language that enables us to describe and model situations, think logically, frame and sustain arguments and communicate ideas with precision. Learners do not know mathematics until they can ‘speak’ it. Effective teaching therefore focuses on the communicative aspects of mathematics by developing oral and written mathematical language.

10. Develops mathematical language through communicative activities

What is to be learned cannot always be stated prior to the learning experience. After a learning event, however, it is important to reflect on the learning that has taken place, making this as explicit and memorable as possible. Effective teachers will also reflect on the ways in which learning has taken place, so that learners develop their own capacity to learn.

11. Recognises both what has been learned and also how it has been learned

Teaching is more effective when it develops mathematical language through communicative activities.



We invited participants to describe any principles that they believe are commonly held, but which are positively unhelpful to teachers.

At the initial conference and the subsequent colloquia, participants were invited to respond to the two open questions:

Participants generated the following list of what they considered to be unhelpful advice:

• • • • • • • • • • • • •

What are the major obstacles to progress? How do these obstacles function? These questions unlocked some strong feelings among participants. The responses are categorised into five overlapping themes, below. It must be emphasised that these themes were not put to participants, they emerged from the responses.

Learn how to do it first – understanding will come later. Repetition will improve understanding. There is a ‘best way’ to teach, an ‘optimal sequence’ for learning, a ‘right way’ to solve each problem. Explain clearly how to do the problem before you give it to your class. Learning must be preceded by instruction. Tell the class your lesson objectives at the beginning of each lesson. Make sure your lesson has three parts. Plan the plenary discussion very carefully and stick to your plan. Cover the syllabus, even if this means hurrying through some parts. Keep emphasising presentation and neatness. Knowing the answer is important. Keep learners busy. Learners go off-task if they talk. Don’t confuse learners by showing them incorrect methods. Use technology whenever you can.

1. Society’s attitude towards mathematics Participants felt that society and its media have a limited, narrow view of mathematics and still promote it as a subject for ‘geeks’. Mathematical incompetence continues to remain socially acceptable. Some also commented on the climate of distrust of teachers and the excessively detailed accountability that permeates and distorts the teaching of mathematics. 2. Teachers’ subject and pedagogical subject knowledge Participants felt that many teachers lack confidence in the subject and an awareness and understanding of appropriate resources (including ICT). They also lack time for continuing professional development. Participants commented that many teachers ‘lack confidence’ in mathematics, ‘can’t see the bigger, interconnected picture’ and ‘can only see one learning pathway through the curriculum’. When unexpected insights, answers or misconceptions arise, for example, many teachers don’t have sufficient subject knowledge to depart from their predetermined plan. This is also evidenced by the overdependence on pre-packaged schemes and textbooks.

Most of these ’unhelpful principles’ are well-intentioned but they become unhelpful when their continued emphasis encourages superficial activity rather than deeper learning.

(B.Vertes)

Mathematics teachers have to overcome society’s attitudes: – that mathematics is “geeky” rather than “cool” – that it is acceptable not to be good at mathematics.

One participant asserted that a common view of mathematics shared by teachers and pupils is that it is a ‘dead weight of predetermined knowledge to be absorbed’, rather than an organic system to explore collaboratively. In attempting to deal with learner disaffection, participants felt that many reward systems encourage extrinsic motivation, rather than an intrinsic satisfaction in the subject itself. They also felt that there is a developing practice of over-praising or praising too soon and teachers being trained to control, not empower.

(P.Williams, H.Pomroy, E.Lawrenson, L.Bell, M.Thomas, M.Elliot)

Compounding this, it was felt that there is a lack of subject-specific CPD within schools and colleges, with opportunities for research and reflection.

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High stakes external testing leads teachers to aim at short-term instrumental understanding.

Teachers’ lack of confidence in mathematics can hinder ‘letting go’ of the content. (P.Drake)

(H.Tanner) This section is concluded with one quote from the adult education sector that illustrates how the above factors can act together to impede learning:

The aim is seen as passing tests rather than learning mathematics.

In adult education settings, the wide range of learner interests and abilities, the likelihood of learners aiming for different qualifications, the sometimes variable attendance patterns, the typical non-resource rich learning environments, all set in the context of funding by results, present challenges to teachers that detract from the core business of igniting learner curiosity, building learner confidence and assuring learner achievement in mathematics.

(P.Griffin)

The publication of examination figures, it was asserted, has lead to short-term initiatives being imposed on teachers, and a reluctance to take risks and innovate. The accompanying excessive paperwork and administration has further detracted from the job of teaching. It should be noted that participants broadly welcomed the new Programmes of Study, with their renewed emphasis on the ‘Key Concepts’ and ‘Key Processes’. These, it was hoped, would have a significant positive influence on the nature of national testing.

(S.Bertenshaw, I.Brazewell, C.Cairns, N.Culverwell

There is a lack of subject specific training in schools. (J.Jones)

Assessment strategies do not reflect effective pedagogy. There is a lack of high-quality, extended CPD opportunities with time for research and learning from reflection.

4. The style and quality of textbooks and other resources

There is a culture of unthinking teacher conformity rather than thoughtful experimentation.

Participants felt that there is still an over-reliance on pre-packaged schemes. Most do not promote the types of learning they value. Participants commented that textbooks, usually associated with awarding bodies, are major obstacles to effective practices, because they atomise the curriculum and promote teaching to the tests. They also commented on teachers' continuing lack of confidence in using ICT.

(P.Griffin)

Fear of testing leads teachers to teach to the test; fear of authority keeps lessons safe and dull.

Some participants expressed a wish to see the access to resources improved, perhaps by the development of a bank of rich tasks. Others however, again commented that teachers have little time to explore new resources with colleagues.

(J.Rickwood)

Teachers have misconceptions about what they are ‘allowed’ to do.

Textbooks discourage connection making. (R.Deakin, S.Wall, M.Colley)

(K.Cross)

Many teachers lack confidence in using ICT for teaching and learning.

5. Initiatives that appear conflicting, disempowering and prescriptive

(R.Deakin, P.Williams, M.Colley)

Participants felt that a simplistic interpretation of the national strategies and the apparent inflexibility and mechanistic nature of inspection regimes (particularly internal ones), can lead to the production of externally acceptable forms of behaviour (e.g. 3-part lessons, learning objectives written on the board before the lesson) and inhibit principled, imaginative teaching.

(K.Jones)

(V.Brown, W.Wilson)

It will be impossible to make meaningful improvements to teaching mathematics if assessment continues to be uninspiring – demanding the regurgitation of facts and ‘teaching to the test’.

3. A taught curriculum defined by assessment Participants felt that assessment should be defined by the curriculum aims, rather than the taught curriculum being defined by assessment. Teaching towards national tests and public examinations has led to an overemphasis on ‘covering content’ in a procedural manner, almost to the exclusion of developing deeper relational understandings and problem-solving strategies. This is at least partly due to the fragmented and artificial nature of many of these assessments. Current formative and summative assessment strategies may thus be said to obstruct effective pedagogies. Participants asserted that it will be impossible to make meaningful improvements to teaching mathematics if assessment continues to be uninspiring and defined by what is measurable rather than what is valued.

Revision books condense content that learners have never had chance to explore before condensing.

Participants commented that a major obstacle has been that teachers adopt local interpretations about what they are ‘allowed’ to do and ‘fear stepping out of line’. This tendency is reinforced by senior managers who lack awareness of effective teaching practices in mathematics. This can result in a culture of unthinking teacher conformity rather than one of thoughtful, creative experimentation.

(P.Lacey).

(H.Isherwood)

It was felt that, whatever the intentions of inspection and/or changes to the strategies, many teachers still believe that they have to teach in a prescribed way and this has led to a feeling of disempowerment which is compounded by frequent changes to national tests and public examinations.

22

23

High stakes external testing leads teachers to aim at short-term instrumental understanding.

Teachers’ lack of confidence in mathematics can hinder ‘letting go’ of the content. (P.Drake)

(H.Tanner) This section is concluded with one quote from the adult education sector that illustrates how the above factors can act together to impede learning:

The aim is seen as passing tests rather than learning mathematics.

In adult education settings, the wide range of learner interests and abilities, the likelihood of learners aiming for different qualifications, the sometimes variable attendance patterns, the typical non-resource rich learning environments, all set in the context of funding by results, present challenges to teachers that detract from the core business of igniting learner curiosity, building learner confidence and assuring learner achievement in mathematics.

(P.Griffin)

The publication of examination figures, it was asserted, has lead to short-term initiatives being imposed on teachers, and a reluctance to take risks and innovate. The accompanying excessive paperwork and administration has further detracted from the job of teaching. It should be noted that participants broadly welcomed the new Programmes of Study, with their renewed emphasis on the ‘Key Concepts’ and ‘Key Processes’. These, it was hoped, would have a significant positive influence on the nature of national testing.

(S.Bertenshaw, I.Brazewell, C.Cairns, N.Culverwell

There is a lack of subject specific training in schools. (J.Jones)

Assessment strategies do not reflect effective pedagogy. There is a lack of high-quality, extended CPD opportunities with time for research and learning from reflection.

4. The style and quality of textbooks and other resources

There is a culture of unthinking teacher conformity rather than thoughtful experimentation.

Participants felt that there is still an over-reliance on pre-packaged schemes. Most do not promote the types of learning they value. Participants commented that textbooks, usually associated with awarding bodies, are major obstacles to effective practices, because they atomise the curriculum and promote teaching to the tests. They also commented on teachers' continuing lack of confidence in using ICT.

(P.Griffin)

Fear of testing leads teachers to teach to the test; fear of authority keeps lessons safe and dull.

Some participants expressed a wish to see the access to resources improved, perhaps by the development of a bank of rich tasks. Others however, again commented that teachers have little time to explore new resources with colleagues.

(J.Rickwood)

Teachers have misconceptions about what they are ‘allowed’ to do.

Textbooks discourage connection making. (R.Deakin, S.Wall, M.Colley)

(K.Cross)

Many teachers lack confidence in using ICT for teaching and learning.

5. Initiatives that appear conflicting, disempowering and prescriptive

(R.Deakin, P.Williams, M.Colley)

Participants felt that a simplistic interpretation of the national strategies and the apparent inflexibility and mechanistic nature of inspection regimes (particularly internal ones), can lead to the production of externally acceptable forms of behaviour (e.g. 3-part lessons, learning objectives written on the board before the lesson) and inhibit principled, imaginative teaching.

(K.Jones)

(V.Brown, W.Wilson)

It will be impossible to make meaningful improvements to teaching mathematics if assessment continues to be uninspiring – demanding the regurgitation of facts and ‘teaching to the test’.

3. A taught curriculum defined by assessment Participants felt that assessment should be defined by the curriculum aims, rather than the taught curriculum being defined by assessment. Teaching towards national tests and public examinations has led to an overemphasis on ‘covering content’ in a procedural manner, almost to the exclusion of developing deeper relational understandings and problem-solving strategies. This is at least partly due to the fragmented and artificial nature of many of these assessments. Current formative and summative assessment strategies may thus be said to obstruct effective pedagogies. Participants asserted that it will be impossible to make meaningful improvements to teaching mathematics if assessment continues to be uninspiring and defined by what is measurable rather than what is valued.

Revision books condense content that learners have never had chance to explore before condensing.

Participants commented that a major obstacle has been that teachers adopt local interpretations about what they are ‘allowed’ to do and ‘fear stepping out of line’. This tendency is reinforced by senior managers who lack awareness of effective teaching practices in mathematics. This can result in a culture of unthinking teacher conformity rather than one of thoughtful, creative experimentation.

(P.Lacey).

(H.Isherwood)

It was felt that, whatever the intentions of inspection and/or changes to the strategies, many teachers still believe that they have to teach in a prescribed way and this has led to a feeling of disempowerment which is compounded by frequent changes to national tests and public examinations.

22

23


Using coaching to encourage risk taking and innovation by teachers.

At the initial conference, in answer to the question, “What practical steps can we take to help ourselves and others to overcome these obstacles?” there were high levels of agreement. Recommendations centred around four broad themes, namely:

(J.Bean, A.Brunt, J.Pattison, R.Sutton)

2. Develop and share experience and resources for learning

1. Improve the provision and quality of professional development opportunities

Participants saw a need for the NCETM and others to collate and exemplify the combined experience, wisdom and expertise of teachers of mathematics, so that these may be shared more widely. A range of resources were suggested, including:

Participants proposed that continuing professional development should become a career-long entitlement and expectation for all teachers. They thought that effective, collaborative professional development should focus on developing participants’ subject and pedagogical knowledge, rather than on the mechanics of implementing the latest ‘initiative’. Professional development should become characterised by reflective teachers researching their own practice and in engaging with the research of others.

• • •

Resources should be found to foster partnerships between institutions, particularly with those in HE. Teachers should be encouraged to develop through extended Masters level courses.

• • • •

Participants said that they needed a wide range of settings in which they can question, enquire, critique their own practices and those of others. Simple practical possibilities include timetabling an hour a week to stop and read a professional journal or explore a website; cultivating a critical friend to develop materials and ideas with; and taking every opportunity to see the practice of others. More ambitious suggestions included the formation of lesson study groups; and groups which meet to develop, trial and report back on new tasks for the classroom.

mathematical tasks and activities that encourage collaborative work and the development of mathematical processes accounts and authentic videos of lessons that exhibit a wide range of effective practices empirical evidence and case studies that describe effective approaches and illustrate their effects on examination results a resource bank of real-life applications of mathematics a glossary of mathematical terms a compendium of proven and successful lessons/tasks teacher discussion areas.

It was recognised that some of these aspects are already being developed on the NCETM portal (e.g. through the discussion boards and the Mathemapedia) and through the Bowland Maths initiative website, but it was also noted that teachers need additional free time to absorb and work with such resources.

Give teachers time to develop ideas of ways of teaching and the resources to help them.

Ensuring that continuous teacher learning is a professional requirement and entitlement for teachers of mathematics in all settings.

(H.Isherwood)

Provide convincing stories to demonstrate that learners can both learn more effectively through active engagement and pass exams.

Creating opportunities for professional development within and across learning settings.

(D.Wright)

3. Use professional standards to inform others about the teaching and learning of mathematics

Use the NCETM exemplification of the professional standards to help teachers and managers understand what is expected of mathematics teachers.

Conference participants recommended that professional standards that describe and exemplify our values, principles and practices are developed and used to educate the perceptions of managers in schools and colleges so that more time may be made available for PD and more informed performance assessment might take place. It should be noted that during the time of the Mathematics Matters project, the NCETM provided some mathematical exemplification of the professional standards for teaching, as well as guidance for managers in using these in performance reviews.

(R.Tanner)

teaching and learning. In particular, they proposed a greater focus on formative assessment by teachers and that the content of summative assessments should be revised to reflect the broader aims of mathematics teaching.

In addition, participants in the project suggested that ways of promoting positive images of mathematics and mathematicians to society in general should continue to be sought – perhaps through sport, media, music, popular role models, family mathematics days, family homeworks, roadshows, etc.

Participants also wanted to see teachers supported to develop professional confidence that good teaching practices can be employed without a detrimental effect on the performance in high status assessment; there exists clear and accessible evidence of good test and examination results as a consequence of not “teaching to the test”.

Participants were clear that we must work to remove common misinterpretations of national guidance, particularly those that emphasise form over substance, such as the “expectation of a 3-part lesson”. Some participants suggested that it would be helpful to provide an authoritative document that would enable teachers to explicitly address these misinterpretations with senior managers. Others suggested using the NCETM exemplification of the professional standards to inform performance management, including professional and career development. It was also recognised that heads of department need support as they fight for the time and resources they need in order to conduct effective planning and CPD.

Somehow downgrade the importance of the exams or at least have broader measures of assessment. (M.Homer)

Define measures of progress that relate to depth rather than only coverage. (M.Dovey

Teachers need to be given permission to do what they know leads to good learning – even if it doesn’t always fit with perceived expectations.

Support/reinstate the culture of sharing which disappeared when schools had to ‘compete’ with each other in league tables.

(R.Hyde)

(J.Edwards) 4. Influence the extent and quality of key stage assessments and public examinations


(A.Brunt, L.McClure, P.Wylie, N.Honey, P.Hough)

As an inclusive community, agree what we want mathematics teaching and learning in classrooms to look like. (G.Wake)

24

Participants expressed many concerns about testing and generally felt the elimination of or at least reduction in the influence of end of key-stage external assessment was desirable, so creating more time for teaching and learning. A number of suggestions were made which participants felt would change the focus away from tests and onto 25


Using coaching to encourage risk taking and innovation by teachers.

At the initial conference, in answer to the question, “What practical steps can we take to help ourselves and others to overcome these obstacles?” there were high levels of agreement. Recommendations centred around four broad themes, namely:

(J.Bean, A.Brunt, J.Pattison, R.Sutton)

2. Develop and share experience and resources for learning

1. Improve the provision and quality of professional development opportunities

Participants saw a need for the NCETM and others to collate and exemplify the combined experience, wisdom and expertise of teachers of mathematics, so that these may be shared more widely. A range of resources were suggested, including:

Participants proposed that continuing professional development should become a career-long entitlement and expectation for all teachers. They thought that effective, collaborative professional development should focus on developing participants’ subject and pedagogical knowledge, rather than on the mechanics of implementing the latest ‘initiative’. Professional development should become characterised by reflective teachers researching their own practice and in engaging with the research of others.

• • •

Resources should be found to foster partnerships between institutions, particularly with those in HE. Teachers should be encouraged to develop through extended Masters level courses.

• • • •

Participants said that they needed a wide range of settings in which they can question, enquire, critique their own practices and those of others. Simple practical possibilities include timetabling an hour a week to stop and read a professional journal or explore a website; cultivating a critical friend to develop materials and ideas with; and taking every opportunity to see the practice of others. More ambitious suggestions included the formation of lesson study groups; and groups which meet to develop, trial and report back on new tasks for the classroom.

mathematical tasks and activities that encourage collaborative work and the development of mathematical processes accounts and authentic videos of lessons that exhibit a wide range of effective practices empirical evidence and case studies that describe effective approaches and illustrate their effects on examination results a resource bank of real-life applications of mathematics a glossary of mathematical terms a compendium of proven and successful lessons/tasks teacher discussion areas.

It was recognised that some of these aspects are already being developed on the NCETM portal (e.g. through the discussion boards and the Mathemapedia) and through the Bowland Maths initiative website, but it was also noted that teachers need additional free time to absorb and work with such resources.

Give teachers time to develop ideas of ways of teaching and the resources to help them.

Ensuring that continuous teacher learning is a professional requirement and entitlement for teachers of mathematics in all settings.

(H.Isherwood)

Provide convincing stories to demonstrate that learners can both learn more effectively through active engagement and pass exams.

Creating opportunities for professional development within and across learning settings.

(D.Wright)

3. Use professional standards to inform others about the teaching and learning of mathematics

Use the NCETM exemplification of the professional standards to help teachers and managers understand what is expected of mathematics teachers.

Conference participants recommended that professional standards that describe and exemplify our values, principles and practices are developed and used to educate the perceptions of managers in schools and colleges so that more time may be made available for PD and more informed performance assessment might take place. It should be noted that during the time of the Mathematics Matters project, the NCETM provided some mathematical exemplification of the professional standards for teaching, as well as guidance for managers in using these in performance reviews.

(R.Tanner)

teaching and learning. In particular, they proposed a greater focus on formative assessment by teachers and that the content of summative assessments should be revised to reflect the broader aims of mathematics teaching.

In addition, participants in the project suggested that ways of promoting positive images of mathematics and mathematicians to society in general should continue to be sought – perhaps through sport, media, music, popular role models, family mathematics days, family homeworks, roadshows, etc.

Participants also wanted to see teachers supported to develop professional confidence that good teaching practices can be employed without a detrimental effect on the performance in high status assessment; there exists clear and accessible evidence of good test and examination results as a consequence of not “teaching to the test”.

Participants were clear that we must work to remove common misinterpretations of national guidance, particularly those that emphasise form over substance, such as the “expectation of a 3-part lesson”. Some participants suggested that it would be helpful to provide an authoritative document that would enable teachers to explicitly address these misinterpretations with senior managers. Others suggested using the NCETM exemplification of the professional standards to inform performance management, including professional and career development. It was also recognised that heads of department need support as they fight for the time and resources they need in order to conduct effective planning and CPD.

Somehow downgrade the importance of the exams or at least have broader measures of assessment. (M.Homer)

Define measures of progress that relate to depth rather than only coverage. (M.Dovey

Teachers need to be given permission to do what they know leads to good learning – even if it doesn’t always fit with perceived expectations.

Support/reinstate the culture of sharing which disappeared when schools had to ‘compete’ with each other in league tables.

(R.Hyde)

(J.Edwards) 4. Influence the extent and quality of key stage assessments and public examinations


(A.Brunt, L.McClure, P.Wylie, N.Honey, P.Hough)

As an inclusive community, agree what we want mathematics teaching and learning in classrooms to look like. (G.Wake)

24

Participants expressed many concerns about testing and generally felt the elimination of or at least reduction in the influence of end of key-stage external assessment was desirable, so creating more time for teaching and learning. A number of suggestions were made which participants felt would change the focus away from tests and onto 25

APPENDICES


References Ahmed, A. (1987). Better Mathematics: A Curriculum Development Study. London: HMSO.

Prompts for Session 1a

Cockcroft, W. H. (1982). Mathematics Counts. London: HMSO.

Practice

You are asked to write a brief report on a memorable and successful lesson Age/ability range

DfES. (2005). Improving Learning in Mathematics. London: Standards Unit, Teaching and Learning Division.

What was the mathematical task?

QCA. (2007). Mathematics: Programmes of Study for Key Stage 3 & 4 and attainment targets. Smith, A. (2004). Making Mathematics Count: The Stationery Office Ltd.

How was the session/task introduced?

Swan, M. (2006). Collaborative Learning in Mathematics: A Challenge to our Beliefs and Practices. London: National Institute for Advanced and Continuing Education (NIACE); National Research and Development Centre for Adult Literacy and Numeracy (NRDC).

How was the session/task sustained?

How was the session/task concluded?

What were the critical moments?

What mathematics was learnt? (on plan and off plan) and what is the evidence of learning?

How was that mathematics learnt?

Other memorable outcomes

Reference to any resources used

26

27

APPENDICES


References Ahmed, A. (1987). Better Mathematics: A Curriculum Development Study. London: HMSO.

Prompts for Session 1a

Cockcroft, W. H. (1982). Mathematics Counts. London: HMSO.

Practice

You are asked to write a brief report on a memorable and successful lesson Age/ability range

DfES. (2005). Improving Learning in Mathematics. London: Standards Unit, Teaching and Learning Division.

What was the mathematical task?

QCA. (2007). Mathematics: Programmes of Study for Key Stage 3 & 4 and attainment targets. Smith, A. (2004). Making Mathematics Count: The Stationery Office Ltd.

How was the session/task introduced?

Swan, M. (2006). Collaborative Learning in Mathematics: A Challenge to our Beliefs and Practices. London: National Institute for Advanced and Continuing Education (NIACE); National Research and Development Centre for Adult Literacy and Numeracy (NRDC).

How was the session/task sustained?

How was the session/task concluded?

What were the critical moments?

What mathematics was learnt? (on plan and off plan) and what is the evidence of learning?

How was that mathematics learnt?

Other memorable outcomes

Reference to any resources used

26

27

Prompts for Session 1b

Principles

Principles for teaching and learning

Ideal and implemented values

Here are some principles that have been emphasised in the recent “Improving Learning in Mathematics” materials.

Write “A” in the appropriate box on each row to show your vision for an ideal mathematics curriculum. Write “B” in the appropriate box on each row to show the values implied by the curriculum that is currently implemented in most schools and other institutions.

Principles used in the Standards Unit material: “Improving Learning in Mathematics” Teaching is more effective when it…

4 = almost all mathematics lessons should contain this aspect 3 = most mathematics lessons should contain this aspect 2 = less than half of mathematics lessons should contain this aspect 1 = few mathematics lessons should contain this aspect More often Types of outcome and types of activity

4

Fluency in recalling facts and performing skills For example: Memorising names and notations Practising routine procedures Interpretations for concepts and representations For example: Discriminating between examples/non-examples Generating representations Constructing relationships Translating between representations Strategies for investigation and problem solving For example: Formulating questions/problems Developing/comparing strategies for solution Monitoring progress Interpreting/evaluating solutions Communicating results Awareness of the nature and values of the educational system For example: Recognising the purposes of learning maths Developing learning/reviewing strategies Knowing what others value

•

builds on the knowledge learners already have


•

exposes and discusses common misconceptions

Learning activities should expose current thinking, create ‘tensions’ by confronting learners with inconsistencies, and allow opportunities for resolution through discussion.

•

uses higher-order questions


•

uses cooperative small group work


•

encourages reasoning rather than ‘answer getting’


•

uses rich, collaborative tasks

The tasks we use should be accessible, extendable, encourage decisionmaking, promote discussion, encourage creativity, encourage ‘what if’ and ‘what if not’ questions.

•

creates connections between topics


•

uses technology in appropriate ways

Computers and interactive whiteboards allow us to present concepts in visual dynamic and exciting ways that motivate learners.

Less often 3

2

1

How far do you agree with these?

What principles would you add?

Appreciation of the power of mathematics in society For example: Appreciate history/cultural foundations Creating/critiquing models of real situations Recognising uses/abuses of maths in society Gaining power over problems in ones own life

Are there principles, commonly believed by teachers that are positively unhelpful for learning mathematics? If so, then what are they?

Please write comments and notes on the back of this sheet.

Now share and discuss your ideas and their implications. 28

29

Prompts for Session 1b

Principles

Principles for teaching and learning

Ideal and implemented values

Here are some principles that have been emphasised in the recent “Improving Learning in Mathematics” materials.

Write “A” in the appropriate box on each row to show your vision for an ideal mathematics curriculum. Write “B” in the appropriate box on each row to show the values implied by the curriculum that is currently implemented in most schools and other institutions.

Principles used in the Standards Unit material: “Improving Learning in Mathematics” Teaching is more effective when it…

4 = almost all mathematics lessons should contain this aspect 3 = most mathematics lessons should contain this aspect 2 = less than half of mathematics lessons should contain this aspect 1 = few mathematics lessons should contain this aspect More often Types of outcome and types of activity

4

Fluency in recalling facts and performing skills For example: Memorising names and notations Practising routine procedures Interpretations for concepts and representations For example: Discriminating between examples/non-examples Generating representations Constructing relationships Translating between representations Strategies for investigation and problem solving For example: Formulating questions/problems Developing/comparing strategies for solution Monitoring progress Interpreting/evaluating solutions Communicating results Awareness of the nature and values of the educational system For example: Recognising the purposes of learning maths Developing learning/reviewing strategies Knowing what others value

•

builds on the knowledge learners already have


•

exposes and discusses common misconceptions

Learning activities should expose current thinking, create ‘tensions’ by confronting learners with inconsistencies, and allow opportunities for resolution through discussion.

•

uses higher-order questions


•

uses cooperative small group work


•

encourages reasoning rather than ‘answer getting’


•

uses rich, collaborative tasks

The tasks we use should be accessible, extendable, encourage decisionmaking, promote discussion, encourage creativity, encourage ‘what if’ and ‘what if not’ questions.

•

creates connections between topics


•

uses technology in appropriate ways

Computers and interactive whiteboards allow us to present concepts in visual dynamic and exciting ways that motivate learners.

Less often 3

2

1

How far do you agree with these?

What principles would you add?

Appreciation of the power of mathematics in society For example: Appreciate history/cultural foundations Creating/critiquing models of real situations Recognising uses/abuses of maths in society Gaining power over problems in ones own life

Are there principles, commonly believed by teachers that are positively unhelpful for learning mathematics? If so, then what are they?

Please write comments and notes on the back of this sheet.

Now share and discuss your ideas and their implications. 28

29

Prompts for Session 1c

Response Sheet for Session 1c

Linking Practice to Principles

Linking values and Principles to Practice

In groups of 3 with members A, B & C

My lesson/task report illustrates the following value(s) and principle(s) in practice: (please tick the appropriate box(es) in each table)

• • • • • •

A spends about 5 minutes describing the lesson reported in Session 1a. B & C listen and comment and question in the light of the values and principles discussed in Session 1b. B may take notes.

VALUES Fluency in recalling facts and performing skills

A, B and C identify which values were being exemplified in the presented example. This is recorded on the sheet below and attached to the example. A becomes B, B becomes C and C becomes A, and the process is repeated twice.

In the light of the discussion participants may revise their lesson/task report or select an alternative.

Interpretations for concepts and representations Strategies for investigation and problem solving Awareness of the nature and values of the educational system Appreciation of the power of mathematics in society

How does the account exemplify this/these value(s)?

PRINCIPLES Teaching is more effective when it… builds on the knowledge learners already have exposes and discusses common misconceptions uses higher-order questions uses cooperative small group work encourages reasoning rather than ‘answer getting’ uses rich, collaborative tasks creates connections between topics uses technology in appropriate ways

How does the account exemplify this/these principle(s)?

30

31

Prompts for Session 1c

Response Sheet for Session 1c

Linking Practice to Principles

Linking values and Principles to Practice

In groups of 3 with members A, B & C

My lesson/task report illustrates the following value(s) and principle(s) in practice: (please tick the appropriate box(es) in each table)

• • • • • •

A spends about 5 minutes describing the lesson reported in Session 1a. B & C listen and comment and question in the light of the values and principles discussed in Session 1b. B may take notes.

VALUES Fluency in recalling facts and performing skills

A, B and C identify which values were being exemplified in the presented example. This is recorded on the sheet below and attached to the example. A becomes B, B becomes C and C becomes A, and the process is repeated twice.

In the light of the discussion participants may revise their lesson/task report or select an alternative.

Interpretations for concepts and representations Strategies for investigation and problem solving Awareness of the nature and values of the educational system Appreciation of the power of mathematics in society

How does the account exemplify this/these value(s)?

PRINCIPLES Teaching is more effective when it… builds on the knowledge learners already have exposes and discusses common misconceptions uses higher-order questions uses cooperative small group work encourages reasoning rather than ‘answer getting’ uses rich, collaborative tasks creates connections between topics uses technology in appropriate ways

How does the account exemplify this/these principle(s)?

30

31

Prompts for Session 2


Identifying & Overcoming Obstacles

A Personal response

What are the main factors that inhibit or modify practice so that it becomes less than effective?

Please write here any comments you have on the nature and structure of today’s colloquium.

(a) What are the major obstacles to progress? How do these obstacles function?

Please write here how you might continue and develop further some of the work started today – individually and/or back at your place of work.

Please write here how you might continue and develop further some of the work started today – with other colleagues in the region who have attended this colloquium. (b What practical steps can we take to help ourselves and others to overcome these obstacles?

Please write here how you might continue and develop further some of the work started today – with other colleagues nationally by using the Centre’s portal.

32

33



Identifying & Overcoming Obstacles

A Personal response

What are the main factors that inhibit or modify practice so that it becomes less than effective?

Please write here any comments you have on the nature and structure of today’s colloquium.

(a) What are the major obstacles to progress? How do these obstacles function?

Please write here how you might continue and develop further some of the work started today – individually and/or back at your place of work.

Please write here how you might continue and develop further some of the work started today – with other colleagues in the region who have attended this colloquium. (b What practical steps can we take to help ourselves and others to overcome these obstacles?

Please write here how you might continue and develop further some of the work started today – with other colleagues nationally by using the Centre’s portal.

32

33

APPENDIX 2

List of participants Initial London Conference Afzal Ahmed Patricia Alexander

Colloquia

The Mathematics Centre, University of Chichester Goldsmiths College – University of London

Colin Matthews

The NCETM

1. Leicester

Dave Miller

AMET (Association of Mathematics Education Teachers) & Keele University

Barry Ablitt

Bishop Grosseteste University College Lincoln

Debbie Morgan

University of Northampton

Doreen Connor Alan Cossins Nikki Cruickshank Liz Durham Michael Elliott Peter Hough Helen Morrell Ray Sutton Maggie Thomas Ray Twigg Frances Watson Jan Watson

Nottingham Trent University Vocational Mathematics Project Bridgewater School Southend Adult Community College St Paul’s Catholic School The NCETM Kimbolton School The NCETM Northampton College HMP Birmingham (Derby College) Gamlingay Village College School of Education, University of Leicester

Barbara Allen

Centre for Mathematics Education, The Open University

Bernard Murphy

MEI

Andrew Noyes

University of Nottingham

Bob Ansell


Adrian Oldknow

The Mathematical Association

Jenni Back

Middlesex University

Mark Patmore

Nottingham Trent University

David Baker

Institute of Education, University of London

Jennie Pennant

BEAM Education

Sue Pope

Qualifications and Curriculum Authority

Chris Belsom

Ampleforth College

Roger Porkess

Mathematics in Education and Industry

Carolyn Brooks

Anglia Ruskin University

Dave Pratt

Lynn Churchman

National Mathematics Partnership


Alison Clark-Wilson

University of Chichester

David Reynolds

University of Plymouth

Julia Croft

University of Bedfordshire

John Rickwood

Burlington Junior School

Kath Cross

Retired HMI

Dan Robinson

John Dabell

Consultant

Specialist Schools and Academies Trust

Els De Geest

University of Oxford

Sue Sanders

University of Swansea

Pat Drake

University of Sussex

Malcolm Swan


Stella Dudzic


Ruth Swinton

Mathematical Association

Julie-Ann Edwards

University of Southampton

Ruth Tanner

Lodge Park Technology College

Steve Feller

Edge Hill University

Howard Tanner

Swansea School of Education

Tony Gardiner

University of Birmingham

Andrew Taylor

AQA

Joy Garvey

Royal Borough of Kingston on Thames

Lin Taylor

St Martin's College

Karen Gladwin


Sidney Tyrrell

Coventry University

Jenny Golding


Geoff Wake

University of Manchester

Richard Goodman

University of Brighton

Linton Waters

Secondary National Strategy for school improvement

4. Newcastle

Joan Ashley Steffi Bludovsky Sue Briggs Pete Griffin Chris Haynes Norma Honey James Knightbridge Keith Noble Lee Northern Richard Perring Jim Thorpe

Liz Bell Shirley Bertenshaw Ian Brazewell Caroline Cairns Rachel Coombes Nigel Culverwell Dan Curran Chris Dallow Richard Deakin Sue Evans Damian Griffith Sue Harkness Liz Henning Teresa Kent Rob Kinnersley Anna Kirwan Emma Lawrenson Barbara Masters Ken McKelvie Janet Pass Hannah Pomroy Ian Porteus Rob Simpson Susan Wall Pam Wignall Pauline Williams Will Wilson

The NCETM Colyton Grammar School The Castle School,Taunton The NCETM DCS The NCETM Blandford School, Dorset Poole Grammar School Cornwall’s Children’s Services Authority DCS The NCETM

Edexcel

Rose Griffiths

University of Leicester School of Education

Anne Watson


3. Canterbury

Linda Haggarty

The Open University

David Wright

Newcastle University

Matt Homer

School of Education, University of Leeds

Ros Hyde


Fahrat Ibrahim

St Martin’s College

Holly Isherwood

The Royal Institution

Rosa Archer Fiona Chapman John Dore Graham Griffiths Sue Houston Andrea Kite

Jane Imrie

The NCETM

St Mary’s University College The Folkestone School for Girls University of Kent LLU/ London South Bank University Hastings College Archers Court Maths and Computing College

Keith Jones


Jane Jones

Specialist Adviser for Mathematics

Vinay Kathotia

Clothworkers’ Fellow in Mathematics – The Royal Institution of Great Britain

Lisa Kirby

CEA@Islington

Peter Lacey

ECARDA Ltd.

Michael Ling

The Royal Statistical Society

Tony Mann

University of Greenwich

Elaine Lambert Snezana Lawrence Pauline Noble Joan Ostheimer Cheryl Periton Margaret Sangster Paula Stone Bob Vertes

Orpington College St Edmund’s Catholic School Hartsdown Technology College Mid Kent College The NCETM Canterbury Christ Church University Canterbury Christ Church University St Mary University

Sarah Mann

ECARDA Ltd.

David Martin

NANAMIC

John Mason

The Open University

The NCETM Middlesbrough LA Heaton Manor School Durham County Council Northumberland County Council Durham University The NCETM Darlington LA Durham LA North Tyneside Council North Tyneside Council

5. Manchester

2. Bath

Jeffrey Goodwin

34

Viv Brown Margaret Colley Chris Cook Shirley Fall Alasdair Findlayson Tony Harries Steve Humble Julia Gibb Brian Hill Wendy Truscott Barbara Thompson

Sherburn High School Bolton Community College Manchester Adult Education Service Trafford College Craven College, Skipton Manchester Adult Education Service Norton College Worcester Sixth Form College Failsworth School Lakes College Failsworth School Liverpool LA The NCETM Wilberforce College Shrewsbury Sixth Form College Stockport College Myerscough Skills Tutor The Sheffield College University of Liverpool Stockport Continuing College Skipton Girls High School University of Liverpool South Hunsley School The NCETM West Cheshire College Ballakermeen High School Salford LA

6. Peterborough John Bean Alison Brunt Marion Dovey Christine Hines Lynne McClure Julie Pattison Pamela Wylie 35

PFEG The NCETM Selby College Leeds Metropolitan University Primary Project Box North Yorkshire County Council Consultant

APPENDIX 2

List of participants Initial London Conference Afzal Ahmed Patricia Alexander

Colloquia

The Mathematics Centre, University of Chichester Goldsmiths College – University of London

Colin Matthews

The NCETM

1. Leicester

Dave Miller

AMET (Association of Mathematics Education Teachers) & Keele University

Barry Ablitt

Bishop Grosseteste University College Lincoln

Debbie Morgan


Doreen Connor Alan Cossins Nikki Cruickshank Liz Durham Michael Elliott Peter Hough Helen Morrell Ray Sutton Maggie Thomas Ray Twigg Frances Watson Jan Watson

Nottingham Trent University Vocational Mathematics Project Bridgewater School Southend Adult Community College St Paul’s Catholic School The NCETM Kimbolton School The NCETM Northampton College HMP Birmingham (Derby College) Gamlingay Village College School of Education, University of Leicester

Barbara Allen

Centre for Mathematics Education, The Open University

Bernard Murphy

MEI

Andrew Noyes


Bob Ansell


Adrian Oldknow

The Mathematical Association

Jenni Back

Middlesex University

Mark Patmore

Nottingham Trent University

David Baker


Jennie Pennant

BEAM Education

Sue Pope

Qualifications and Curriculum Authority

Chris Belsom

Ampleforth College

Roger Porkess


Carolyn Brooks


Dave Pratt

Lynn Churchman

National Mathematics Partnership


Alison Clark-Wilson

University of Chichester

David Reynolds

University of Plymouth

Julia Croft

University of Bedfordshire

John Rickwood

Burlington Junior School

Kath Cross

Retired HMI

Dan Robinson

John Dabell

Consultant

Specialist Schools and Academies Trust

Els De Geest


Sue Sanders

University of Swansea

Pat Drake

University of Sussex

Malcolm Swan


Stella Dudzic


Ruth Swinton


Julie-Ann Edwards


Ruth Tanner

Lodge Park Technology College

Steve Feller

Edge Hill University

Howard Tanner

Swansea School of Education

Tony Gardiner

University of Birmingham

Andrew Taylor

AQA

Joy Garvey

Royal Borough of Kingston on Thames

Lin Taylor

St Martin's College

Karen Gladwin


Sidney Tyrrell

Coventry University

Jenny Golding


Geoff Wake

University of Manchester

Richard Goodman

University of Brighton

Linton Waters

Secondary National Strategy for school improvement

4. Newcastle

Joan Ashley Steffi Bludovsky Sue Briggs Pete Griffin Chris Haynes Norma Honey James Knightbridge Keith Noble Lee Northern Richard Perring Jim Thorpe

Liz Bell Shirley Bertenshaw Ian Brazewell Caroline Cairns Rachel Coombes Nigel Culverwell Dan Curran Chris Dallow Richard Deakin Sue Evans Damian Griffith Sue Harkness Liz Henning Teresa Kent Rob Kinnersley Anna Kirwan Emma Lawrenson Barbara Masters Ken McKelvie Janet Pass Hannah Pomroy Ian Porteus Rob Simpson Susan Wall Pam Wignall Pauline Williams Will Wilson

The NCETM Colyton Grammar School The Castle School,Taunton The NCETM DCS The NCETM Blandford School, Dorset Poole Grammar School Cornwall’s Children’s Services Authority DCS The NCETM

Edexcel

Rose Griffiths

University of Leicester School of Education

Anne Watson


3. Canterbury

Linda Haggarty

The Open University

David Wright

Newcastle University

Matt Homer

School of Education, University of Leeds

Ros Hyde


Fahrat Ibrahim

St Martin’s College

Holly Isherwood

The Royal Institution

Rosa Archer Fiona Chapman John Dore Graham Griffiths Sue Houston Andrea Kite

Jane Imrie

The NCETM

St Mary’s University College The Folkestone School for Girls University of Kent LLU/ London South Bank University Hastings College Archers Court Maths and Computing College

Keith Jones


Jane Jones

Specialist Adviser for Mathematics

Vinay Kathotia

Clothworkers’ Fellow in Mathematics – The Royal Institution of Great Britain

Lisa Kirby

CEA@Islington

Peter Lacey

ECARDA Ltd.

Michael Ling

The Royal Statistical Society

Tony Mann

University of Greenwich

Elaine Lambert Snezana Lawrence Pauline Noble Joan Ostheimer Cheryl Periton Margaret Sangster Paula Stone Bob Vertes

Orpington College St Edmund’s Catholic School Hartsdown Technology College Mid Kent College The NCETM Canterbury Christ Church University Canterbury Christ Church University St Mary University

Sarah Mann

ECARDA Ltd.

David Martin

NANAMIC

John Mason

The Open University

The NCETM Middlesbrough LA Heaton Manor School Durham County Council Northumberland County Council Durham University The NCETM Darlington LA Durham LA North Tyneside Council North Tyneside Council

5. Manchester

2. Bath

Jeffrey Goodwin

34

Viv Brown Margaret Colley Chris Cook Shirley Fall Alasdair Findlayson Tony Harries Steve Humble Julia Gibb Brian Hill Wendy Truscott Barbara Thompson

Sherburn High School Bolton Community College Manchester Adult Education Service Trafford College Craven College, Skipton Manchester Adult Education Service Norton College Worcester Sixth Form College Failsworth School Lakes College Failsworth School Liverpool LA The NCETM Wilberforce College Shrewsbury Sixth Form College Stockport College Myerscough Skills Tutor The Sheffield College University of Liverpool Stockport Continuing College Skipton Girls High School University of Liverpool South Hunsley School The NCETM West Cheshire College Ballakermeen High School Salford LA

6. Peterborough John Bean Alison Brunt Marion Dovey Christine Hines Lynne McClure Julie Pattison Pamela Wylie 35

PFEG The NCETM Selby College Leeds Metropolitan University Primary Project Box North Yorkshire County Council Consultant

NOTES

36

NOTES

36

Mathematics Matters Final Report

www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching.